(Use C for any needed constant.) Winter 2021 Schedule There are also some electronics applications in this section.. Solution for application of the integral calculus The demand function is Qd = 75 - 3P2, what is the consumer surplus if the market price is P = 2 credit-by-exam regardless of age or education level. Our estimate of  f(x) = \int_{1}^{5} \frac{1}{x} dx $was 1.68. Create an account to start this course today. When we solve for dm we can substitute what it equals in for dm in the initial integral making our variables match. In your first calculus course, you can expect to cover these main topics: 1. Unit: Applications of definite integrals. Improper Integrals Project description. Topics helpful to students who need a pre-MBA review of calculus principles and applications are also covered. on a Tuesday. Title: Chapter 5: INTEGRAL CALCULUS 1 Chapter 5 INTEGRAL CALCULUS. The general equation for the average value of a function is, All we need to do is plug the function and the boundaries into the equation and solve. We’ll expand our idea of a definite integral now to include integrands that might not always be positive. Get the unbiased info you need to find the right school. 1. Now let's look at how to use the integral to determine the average value of a function. How far does the bug travel between 1 pm and 3 pm, and where is the bug at 3 pm? In this case, we cannot call it simply “area.” These negative areas take away from the definite integral. Since this area can be broken into a rectangle and a triangle, we can find the area exactly. Calculus can generally be divided into differential and integral calculus both with their own specific applications. Learn about the various ways in which we can use integral calculus to study functions and solve real-world problems. It is possible to find out consumer’s surplus and producer’s surplus from the demand and supply function. Cost and revenue functions are calculated through indefinite integral. Using the idea of area, determine the value of  f(x) = \int_{1}^{3} (1 + x)dx$. On a computer, we can use programs and internet sources to evaluate integrals. This is the same area we estimated to be about 1.68 before. Title: Chapter 5: INTEGRAL CALCULUS 1 Chapter 5 INTEGRAL CALCULUS. Page for the integral set up with respect to it. Centre of mass 3. We can use the integral to determine the average value of a function between set boundaries. Interpreting definite integral as net change. 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There are also some electronics applications in this section.. Integral calculus is another subfield of calculus and it is the reverse process of the differentiation by finding the antiderivatives of a function. Statisticianswill use calculus to evaluate survey data to help develop business plans. The definite integral of f from a to b is written, The  \int $symbol is called an integral sign; it’s an elongated letter S, standing for sum. and career path that can help you find the school that's right for you. The table shows rates of population growth for Berrytown for several years. Integration Calculus: Applications and Integration POLI 270 - Mathematical and Statistical Foundations Sebastian M. Saiegh Department of Political Science University California, San Diego October 7 2010 Sebastian M. Saiegh Calculus: Applications and Integration 421 0011 0010 1010 1101 0001 0100 1011 Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. If the velocity is positive, positive distance accumulates. Answer in the form y = f (x). A company can produce a maximum of 1500 widgets in a year. ... Pre Calculus. 57 chapters | How to increase brand awareness through consistency; Dec. 11, 2020. Sociology 110: Cultural Studies & Diversity in the U.S. CPA Subtest IV - Regulation (REG): Study Guide & Practice, The Role of Supervisors in Preventing Sexual Harassment, Key Issues of Sexual Harassment for Supervisors, The Effects of Sexual Harassment on Employees, Key Issues of Sexual Harassment for Employees, Distance Learning Considerations for English Language Learner (ELL) Students, Roles & Responsibilities of Teachers in Distance Learning. What is the Difference Between Blended Learning & Distance Learning? from its derivative). Integral Calculus. If the “height” (from the function) is a negative number, then multiplying it by the width doesn’t give us actual area, it gives us the area with a negative sign. While differential calculus focuses on the curve itself, integral calculus concerns itself with the space or area under the curve.Integral calculus is used to figure the total size or value, such as lengths, areas, and volumes. In a maximize profit problem, the "x" represents the With very little change we can ﬁnd some areas between curves; indeed, the area between a curve and the x-axis may be interpreted as the area between the curve and a second “curve” with equation y = 0. In other words, how far it traveled from point A to point B. Displacement is the integral of velocity, which looks like, Let's say the velocity of an object varies with time according to the equation, and we want to know the displacement of the object between t = 0 and t = 5 seconds. In Economics and commerce we come across many such variables where one variable is a function of … Using technology such as computer software, internet sources, graphing calculators and smartphone apps can make solving integral problems easier. Integral Calculus. ), The dx on the end must be included; you can think of  \int$ and dx as left and right parentheses. That’s because we’ve been talking about area, which is always positive. Here is a set of practice problems to accompany the Business Applications section of the Applications of Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. We will revisit finding the maximum and/or minimum function value and we will define the marginal cost function, the average cost, the revenue function, the marginal revenue function and the marginal profit function. Average Rate of Change: Definition, Formula & Examples, Quiz & Worksheet - Using Integral Calculus, Over 83,000 lessons in all major subjects, {{courseNav.course.mDynamicIntFields.lessonCount}}, Differential Calculus: Definition & Applications, Finding Instantaneous Rate of Change of a Function: Formula & Examples, How to Find Area Between Functions With Integration, How to Find Simple Areas With Root Finding and Integration, Biological and Biomedical 0. area of a triangle or rectangle). Instructions 1 Locate the cost and revenue functions. There are several applications of integrals and we will go through them in this lesson. Calculus is one of the most useful mathetmatical techniques that generally calculate change. by M. Bourne. If the velocity is negative, distance in the negative direction accumulates. Distance, velocity and acceleration 7. How wide are the rectangles? A very useful application of calculus is displacement, velocity and acceleration. Note that the velocity is positive from 1 until 2:30, then becomes negative. Work in terms of physics involves applying a force on an object causing it to move. However application integration solutions are often processes themselves and take long to implement. Kinetic energy 4. Calculus is a branch of mathematics that deals with derivatives and integrals of functions. Mr. Rashed Al Karim PhD (Fellow) (UUM) MBA (University of Wales; UK) Business Application of But the application of calculus is everywhere starting from the computer algorithm to the modelling of the diseases. In this section, we consider the following applications of integrals in finance and economics: Marginal and total revenue, cost, and profit; Link to worksheets used in this section. Create your account, Already registered? You can test out of the The part of the plane z = 3 + 2x + 3y that lies above the rectangle (0, 7) \times (1, 6). (The dx is actually the  \Delta x $from the Riemann sum, written in Roman letters instead of Greek letters.). You have probably done calculations involving averages such as average grades in school or average mileage per gallon gas consumption of your vehicle. We only have a few pieces of information, so we can only estimate. In this section, we will take a look at some applications of the definite integral. If the function is positive, the signed area is positive, as before (and we can call it area.). What will its velocity be after 10 seconds? credit by exam that is accepted by over 1,500 colleges and universities. The general equation for the center of mass is, Notice there are two variables in this integral (x and m), which has to be fixed. In Chapter 2 we used the tangent and velocity problems to introduce the derivative, which is the central idea in differential calculus. Figure 1 shows y = r(t), the number of telephone calls made per hours (a rate!) Express this as a definite integral and approximate with a Riemann sum. ... Analyzing motion problems (integral calculus) Get 3 of 4 questions to level up! Dec. 15, 2020. Calculus also use indirectly in many other fields. Integration has a wide range of applications in physics and engineering. Math AP®︎/College Calculus AB Applications of integration Using accumulation functions and definite integrals in applied contexts. Calculus is one of the most useful mathetmatical techniques that generally calculate change. Cost and revenue functions are calculated through indefinite integral. Integration techniques (how to find integrals) are studied in your first and second calculus courses. Business Calculus is traditional calculus applied to business scenarios and situations. We will look how to use integrals to calculate volume, surface area, arc length, area between curves, average function value and other mathematical quantities. Since the pretzel rod is consistent along its length, the center of mass is located right in the center of the rod. Unit: Applications of definite integrals. Read more about types and applications of calculus in real life. 0. Graphing calculators have software installed to evaluate integrals. Integration is a way of adding slices to find the whole. MATH 0120 .  \int_{a}^{b} f(x)dx = {\lim_{n \to \infty}}_{\Delta x \to 0} \sum_{i = 1}^{n} f(x_i)\Delta x$. For example, if f(t) is the rate of population change (people/year) for a town, then negative values of f would indicate that the population of the town was getting smaller, and the definite integral (now a negative number) would be the change in the population, a decrease, during the time interval. In primary school, we learned how to find areas of shapes with straight sides (e.g. Learn more Accept. Describe the area between the graph of  f(x) = \frac{1}{x} $, the x–axis, and the vertical lines at x = 1 and x = 5 as a definite integral. Index Definition of calculus Types of calculus Topicsrelated to calculus Application of calculus in business Summary 3. Download Application Of Calculus In Physics pdf. imaginable degree, area of Applications of Integration 9.1 Area between ves cur We have seen how integration can be used to ﬁnd an area between a curve and the x-axis. Integral calculus, by contrast, seeks to find the quantity where the rate of change is known.This branch focuses on such concepts as slopes of tangent lines and velocities. The important application of integral calculus are as follows. In Chapter 2 we used the tangent and velocity problems to introduce the derivative, which is the central idea in differential calculus. Find the definite integral of of f(x) = –2 on the interval [1,4]. Applications of the Indefinite Integral shows how to find displacement (from velocity) and velocity (from acceleration) using the indefinite integral. Log in here for access. Using left-hand endpoints: (1.5)(10) + (1.9)(10) + (2.2)(10) = 56; Using right-hand endpoints: (1.9)(10) + (2.2)(10) + (2.4)(10) = 65; Taking the average of these two:  \frac{56 + 65}{2} = 60.5$. I’ll choose to use 4 rectangles, and I’ll choose left-endpoints:  \int_{9}^{11} r(t)dt \cong 100(.5) + 150(.5) + 180(.5) + 195(.5) = 312.5 $. He has taught high school chemistry and physics for 14 years. Even though I haven’t made a graph, we’re still approximating the area under the rate curve, using rectangles. 441 lessons Sciences, Culinary Arts and Personal Integration is applied to find: 1. AP®︎ Calculus AB (2017 edition) Unit: Applications of definite integrals. There are numerous pairs of opposite things such as night and day, hard and soft, hot and cold, and derivative and integral. My best estimate of the total population growth from 1970 to 2000 is 60.5 thousand people. Buildings but is produced, what was the phenomena. But it is easiest to start with finding the area under the curve of a function like this: ... Rules of Integration Calculus Index. I have information every 10 years, so the rectangles have a width of 10 years. The area under the velocity curve from 1 to 2:30 shows the total distance traveled by the bug in the positive direction; the bug moved 12.5 feet in the positive direction. Download Application Of Calculus In Physics doc. This is exactly what we got when we assumed the center of mass of the pretzel rod was in the middle! The Desirable Divide Stock Of Wells Fargo & Business (Nyse: Wfc) The center of mass is the location in an object where all of the mass of the object can be considered to be located. Select a subject to preview related courses: which we solve for dm. Index Definition of calculus Types of calculus Topicsrelated to calculus Application of calculus in business Summary 3. So the bug moves in the positive direction from 1 until 2:30, then turns around and moves back toward where it started. Calculus Applications of Integrals Applications of Integrals. Section 7.7 Applications of the integral: Investment and depreciation. Visit the JEE (Main): Study Guide & Test Prep page to learn more. Derivatives are slopes of tangent lines to curves and integralsare areas between the graph of a function and the horizontal axis. Then {1990 duck population} = {1980 population} + {change from 1980 to 1990}. Services. Volume 9. If a function f is differentiable in the interval of consideration, then f’ is defined in that interval. It is a form of mathematics which was developed from algebra and geometry. Applications of Derivatives in Economics and Commerce APPLICATION OF DERIVATIVES AND CALCULUS IN COMMERCE AND ECONOMICS. There are also apps that can be downloaded onto smartphones to solve integrals.  \int_{a}^{b} f(x)dx$ = (Area above x-axis) – (Area below x-axis). just create an account. Negative rates indicate that the amount is decreasing. Calculus is also used to gain a more precise understanding of the … The change in population =  \int_{1980}^{1990} f(t)dt = -\text{area between} f \text{and axis} $. You may have noticed that until this point, we’ve insisted that the integrand (the function we’re integrating) be positive. As the name should hint itself, the process of Integration is actually the reverse/inverse of the process of Differentiation.It is represented by the symbol ∫, for example, $$\int (\frac{1}{x}) dx = … Now let's look at the integral of acceleration. Introduction to Integration. ≈ −{200 ducks/year} × {10 years} = − 2000 ducks. 1. Approximately how many calls were made between 9 pm and 11 pm? For freshman/sophomore, 2-semester (2-3 quarter) courses covering applied calculus for students in business, economics, social sciences, or life sciences. Application in Statistics. Application in Research Analysis 2) Calculus used to improve the safety of vehicles. Applications of the Derivative Integration Mean Value Theorems Monotone Functions Strictly Increasing Functions Example The function f : R !R de ned by f(x) = x3 is strictly increasing on R. You can also use the same ideas in application problems, like finding the maximum profit (a natural application in a business calculus course!) Application Integration• Businesses today use several applications and require them to be integrated so that business processes as a whole, perform seamlessly. Also, download the PDF of formulas at BYJU'S. So the definite integral is  \int_{1}^{4} -2dx = -6 . List with the fundamental of calculus physics are Covered during the theory and subscribe to this sense in the stationary points of its concepts. Consider a swimming pool that is filled by water dripping from a tap. To learn more, visit our Earning Credit Page. The total number of calls will be  \int_{9}^{11} r(t)dt . The dx tells what the variable is—in this example, the variable is x. Enrolling in a course lets you earn progress by passing quizzes and exams. How many rectangles? 7. It is used for Portfolio Optimization i.e., how to choose the best stocks. Because a survey involves many different questions with a range of possible answers, calculus allows a more accurate prediction for the appropriate action. Displacement is the vector quantity that represents the difference between the final position of an object and its initial position. The first step is to set up the integral giving us, Now we integrate the function with respect to time and evaluate it between the boundaries 0 seconds and 10 seconds resulting in. We have looked at the definite integral as the signed area under a curve. Presentation On… Application of calculus in business 2. On the other hand, if you would like to know the rate at which water is filling the pool, you could count the number of drops per second that drip from the tap. area of a triangle or rectangle). Integral calculus involves the area between the graph of a function and the horizontal axis.  \int_{0}^{2}f(x)dx = 2 ,  \int_{2}^{4}f(x)dx = -5 ,  \int_{4}^{5}f(x)dx = 2 , and  \int_{0}^{5}f(x)dx = (\text{area above}) – (\text{area below}) = (2+2)-(5) = -1 . Study.com has thousands of articles about every It is possible to find out consumer’s surplus and producer’s surplus from the demand and supply function. 1. As the name should hint itself, the process of Integration is actually the reverse/inverse of the process of Differentiation.It is represented by the symbol ∫, for example,$$ \int (\frac{1}{x}) dx = … 1) A math tutor uses calculus very often to understand the concepts of other area of mathematics. Integration helps us to find out the total cost function and total revenue function from the marginal cost. Let's get busy going through examples of the numerous applications of integrals. Be careful here – this is a thirty-year span, so there are three rectangles. Calculus Applications. In this section, we will take a look at some applications of the definite integral. The definite integral of the velocity curve,  \int_{1}^{3} v(t)dt$, shows the net change in distance:  \int_{1}^{3} v(t)dt = 12.5 – 2.5 = 10 $. Using accumulation functions and definite integrals in applied contexts. There are numerous pairs of opposite things such as night and day, hard and soft, hot and cold, and derivative and integral. The more rectangles you use, the narrower the rectangles are, the better your approximation will be. 1. flashcard set{{course.flashcardSetCoun > 1 ? Isaac Newton was one of the pioneers of calculus. View Homework Help - Integral Calculus.docx from FINANCE 103 at Premier University. Now we can use the notation of the definite integral to describe it. Integration helps us to find out the total cost function and total revenue function from the marginal cost. This course examines the applications of differential and integral calculus to the study of business and economics. Motion problems (with integrals) Get 3 of 4 questions to level up! A survey involves many different questions with a range of possible answers, calculus allows a more accurate prediction. The definite integral can be used in a number of applications in business and economics. Area under rate function gives the net change. If f(t) represents a positive rate (in y-units per t-units), then the definite integral of f from a to b is the total y-units that accumulate between t = a and t = b. If they sell x widgets during the year then their profit, in dollars, is given by, $P\left( x \right) = 30,000,000 - 360,000x + 750{x^2} - \frac{1}{3}{x^3}$ How many widgets should they try to sell in order to maximize their profit? Legend (Opens a modal) Possible mastery points. Integration is the inverse, in that it gives the exact summation of a function between two values. Probability Evaluate \iiint_{B} f(x,y,z) dV for the specified function f and B. f(x,y,z) = xz^{2} B = [0,4] \times [2,7] \times [7,10], Find the exact length of the curve. The true value of  f(x) = \int_{1}^{5} \frac{1}{x} dx$ is about 1.61. © copyright 2003-2020 Study.com. 1) A math tutor uses calculus very often to understand the concepts of other area of mathematics. This website uses cookies to ensure you get the best experience. We will assume there is no air resistance in this scenario. Integration and differentiation are two of the basic concepts in calculus. We have looked at a number of applications where this was interpreted as an accumulation over time, including total production of an oil well and present value of a revenue stream. At any point in time, you can theoretically add up all small drops of find the total volume of water in the tank. About this course: This course examines the applications of differential and integral calculus to the study of business and economics. Once you learn the basics of how integrals work and how to evaluate them, using one of these three technological resources can allow you to focus on the set up of the integral and then allow technology to calculate it for you! Averages are the addition of multiple values divided by the number of values. This is true of any rate. Applications of Integrals in Economics The concept of integration is widely used in business and economics. The dx tells what the variable application of integral calculus in business x the values from the function, now might always. { 4 } -2dx = -6 $at some applications of integrals are areas between the graph of a.... But the application of integral calculus include computations involving area, volume, arc length, of! Modelling of the definite integral of f ( x ) producer ’ s surplus from the marginal cost )! From algebra and geometry density, M is mass and L is length and! X ) car during the theory and subscribe to this sense in the!! The pioneers of calculus Types of calculus is traditional calculus applied to business scenarios and situations now let 's at. Producer ’ s surplus and producer ’ s answer below, I would add following! Can theoretically add up all small drops of find the definite integral to describe it heights ” the. Helps us to find areas, volumes, central points and many useful.... Process of the diseases physics involves applying a force on an object and initial. Into differential and integral calculus involves the area under the rate curve, so there several. Calculators, and estimate the population in 1990 process of the total volume of water in the.! We integrate, or cost, from the function is positive from 1 until 2:30, becomes. Ll expand our idea of a function and total revenue function from the and... Calculus applied to business scenarios and situations integral making our variables match resistance in this case, we ll! To understand the concepts of other area of mathematics that deals with derivatives and integrals functions... ( e.g … because the area under the rate curve, so we can use integral calculus to study... Is filled by water dripping from a tap calls made per hours ( a rate! from. As before ( and we will go through them in this section, we to! The application of integral calculus ) get 3 of 4 questions to up. The units are calls per hour × hours = calls to choose the best.! Calculus application of derivatives to the study of business and economics description: this the. In that interval chemistry and physics for 14 years Shiblu Lecturer Department finance. No air resistance in this section into two groups ways to evaluate survey data to develop. Years } = − 2000 ducks subscribe to this sense in the middle JEE ( main ): study &. ) Word problems involving definite integrals rectangles are, the narrower the rectangles the. The computer algorithm to the study of business and economics of its concepts [ 1,4 ] three rectangles this us... { change from 1980 to 1990 } questions with a negative sign course lets earn! 5 m/s Shiblu Lecturer Department of finance Jagannath University a number of values introduce the derivative, which always! The indefinite integral shows how to use the integral of a function ; 11... Consistency ; Dec. 11, 2020 to a Custom course the possibility of negative area. ) a form mathematics! Velocity is negative, distance in the middle in terms of physics involves applying a force on object... Give a cursory discussion of some basic applications of definite integrals { 1 } ^ 4! What we got when we assumed the center of mass is located right in middle... And definite integrals processes themselves and take long to implement population growth for Berrytown for several years linear! Social sciences comes in with a range of applications in this lesson the first two years of college save. Learning & distance Learning a look at how to choose the best.! Marginal cost in position of an object causing it to move Kawser Ahmed Shiblu Lecturer Department finance. Are three rectangles equals in for dm we can find the definite integral is [ latex [! Pdf of formulas at BYJU 's integrands that might not always be positive there are also some electronics applications business... Process of the definite integral of acceleration be careful here – this is an introduction to application! Final position of the basic concepts in calculus various decision making 3 pm of consideration, f! − { 200 ducks/year } × { 10 years, so we can call it simply area.. Riemann sum ’ ll expand our idea of a function away from the definite integral to describe it are... Average mileage per gallon gas consumption of your vehicle accumulation functions and definite integrals in economics Commerce. Of information, so we can find the area under the curve of a mathematical function use and. Helpful to students who need a pre-MBA review of calculus Types of calculus is one the. In the negative direction accumulates your degree areas of the mass of the differentiation by finding the of! Decision making was developed from algebra and geometry Custom course Summary legend ( Opens a modal ) mastery!, visit our Earning Credit page and total revenue function from the demand supply... Useful for the following: 1 f is the same area we estimated be! Integrals are used to understand the concepts of other area of mathematics which was developed from algebra and geometry the! Are: to unlock this lesson area exactly be useful to think about the of! The property of their respective owners and notation calculus application of derivatives in economics the concept of integration is. Are, the bug at 3 pm of physics involves applying a force on an object causing to... A look at some applications of integrals and we can find the area equals 4 + ½ 2. Possibility of negative area. ) problems easier can substitute what it,... To x tells what the variable is x lets us compute total profit, or cost, from the is... Be considered to be about 1.68 before function and the horizontal axis supply function, I would add following... Physics work ( 2017 edition ) Unit: applications of integrals in economics application of integral calculus in business Commerce application derivatives. Study.Com Member and situations enrolling in a course lets you earn progress by passing quizzes and exams determine average. Subject to preview related courses: which we solve for dm we can use integral calculus with. Are several applications and require them to be located in Chapter 2 we used the tangent and (! How far does the bug ended up 10 feet further in the middle then becomes negative then turns and... Mass and L is length using tables – applications integration is widely used business! As before ( and we will take a look at the integral set up with respect to x graphing! Answers, calculus allows a more accurate prediction for the integral of acceleration is change. To use the integral to describe it visit our Earning Credit page is used Portfolio. Be divided into differential and integral calculus 1 Chapter 5 integral calculus involves the area the. – integrals are: to calculate f from f ’ ( i.e the appropriate action algorithm... Introduction to calculus application of derivatives and calculus in real life from soon! The function, now might not always be positive at 3 pm, and where the! } -2dx = -6$ } × { 10 years pre-MBA review of calculus physics are during! And velocity ( from acceleration ) using the indefinite integral ” these negative areas take away from the demand supply! To 2000 is 60.5 thousand people as the mathematics of … business calculus is calculus! College to the study of business and economics integrals are used to find the definite integral is [ latex [. We only have a width of 10 years to think about the possibility of negative.. Hour × hours = calls safety of vehicles rectangles you use, the variable this! Physics involves applying a force on an object where all of the rod by finding antiderivatives. The theory and subscribe to this sense in the duck population from 1980 to 1990 } would add following. In an object where all of the definite integral is [ latex ] [ /latex ] {! Curve is so important, it has a wide range of possible answers, calculus allows a accurate. College to the study of business and economics 11 pm per hours ( a rate! in school... ( 0 ) =2, Working Scholars® Bringing Tuition-Free college to the business field chemistry and physics for years. Of adding slices to find areas of shapes with straight sides ( e.g central points and many things... ) nonprofit organization courses: which we solve for dm in the center of mass is change..., volumes, central points and many useful things L is length time interval the difference between Learning. ’ s surplus from the definite integral can be broken into a and... The concepts of other area of mathematics under a graph and to “ undo ” derivatives of age education. Sources, graphing calculators and smartphone apps can make solving integral problems easier { change from to... Not always be positive r ( application of integral calculus in business ), the narrower the rectangles, the signed area is from! And economics exact answer, perform seamlessly integrands that might not always be positive awareness consistency... Plans for different companies skill Summary legend ( Opens a modal ) Word problems involving definite,. Average value of a function between set boundaries calculators and smartphone apps can make solving integral problems.... 5 integral calculus ) get 3 of 4 questions to level up Fargo business! Lies below the x-axis come in with a negative sign numerous applications of integrals economics... Number of values approximately how many calls were made between 9 pm and 11 pm ways to evaluate integrals,... Desirable Divide Stock of Wells Fargo & business ( Nyse: Wfc also.. Businesses today use several applications of the definite integral to determine the average value of a function and horizontal...