Evaluation Theorem

4 Key excerpts on "Evaluation Theorem"

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  eBook - ePub

  CLEP® Calculus Book + Online

  • Gregory Hill, Mel Friedman(Authors)
  • 2013(Publication Date)
  • Research & Education Association

    (Publisher)

  b are positive or negative.

  The left base is 2a – 1 and the right base is 2b – 1. The height is b – a.

  The sum of the two results is b3 – a3 + b2 – a2 – b + a.

  6.6FUNDAMENTAL THEOREM OF CALCULUS

  One of the most impressive developments in calculus was the connection between the differential and integral branches found in what came to be called the Fundamental Theorem of Calculus. One part of the Fundamental Theorem asserts that a definite integral of a continuous function is a function of its upper limit and is therefore itself differentiable. A second part of the Fundamental Theorem establishes how to analytically evaluate a definite integral. Both parts will be presented here without formal proof, but if it enhances your understanding, any online search for the proof of the Fundamental Theorem of Calculus will produce an abundance of sources to read and study.

  Fundamental Theorem of Calculus (Part 1)

  If f is a continuous function, a is a constant, and then

  This theorem essentially says, “The instantaneous rate of change of accumulation, or loss, of area between a function and the x-axis at any point, is equal to the function value of the integrand at that point.” Let’s take an intuitive way to explain what is happening, using a discrete approach to the concept of accumulating area under a curve. Think of a definite integral as a Riemann sum over a given interval. As the upper limit of the given interval changes, accumulated area increases or decreases by “adding” another infinitely thin rectangle to the previously summed rectangles. If the function values are “large” at a particular x value, then the next rectangle added will add more area than if the function values are small. It follows logically that the amount of change in accumulated area is related to the magnitude of the function being integrated.

  Notice that the upper limit is simply x. How would this part of the theorem change if the upper limit was some function of x, say h(x

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  eBook - ePub

  Mathematical Economics

  • Arsen Melkumian(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  9    Indefinite and definite integrals

  Integrals play a twofold role in calculus. The so-called indefinite integral is an operation inverse to differentiation: the integral of function f (x) is a function whose derivative gives us f (x). We have seen the importance of derivatives throughout the textbook, and naturally the inverse operation also plays an important role in mathematical and economic analysis. Later, in Chapter 12 , we study relations between quantities and their rates of change – the so-called differential equations. Solving differential equations is impossible without indefinite integrals.

  The other type of integral is the definite integral. Geometrically a definite integral represents the area under a curve. But in applications the meaning of integration is totaling continuous quantities. For instance, to reconstruct profit from its rate of change we would integrate the latter function. Another application of integrals is in studying consumer–producer surplus.

  9.1 Indefinite integrals

  If f (x) and F(x) are some functions of x such that

  then F(x) is called an antiderivative of f (x).

  For example, the function F1 (x) = 3x2 + 5 is an antiderivative of f (x) = 6x. Note that F1 (x) = 3x2 + 5 is not the only antiderivative of f (x). In fact, the function f (x) = 6x has infinitely many antiderivatives and all antiderivatives of f (x) are functions of the form

  where C is a constant. We will refer to F(x) = 3x2 + C as the general antiderivative of f (x) and we will write

  to indicate that. The symbol is referred to as the integral sign, 6x is the integrand and C is the constant of integration. The expression is known as the indefinite integral of f (x) = 6x.

  THEOREM 9.1: Let f (x) be a differentiable function of x, and k, n and C be some constants. Then

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  eBook - ePub

  AP® Calculus AB/BC All Access Book + Online + Mobile

  • Stu Schwartz(Author)
  • 2013(Publication Date)
  • Research & Education Association

    (Publisher)

  u clearly doesn’t work here. The only way to do this is to recognize its form from the integration rules above.

  TEST

  TIP

  Integration is practice, practice, practice and many times, trial and error. Hopefully your calculus course has done a lot of drilling on integration techniques. You will certainly be tested on it on the AP exam. If there is a tough problem that appears in the multiple-choice section of the exam that you cannot do, you can always take the derivative of each of your 5 answers and see which one is the original expression. It is potentially time-consuming, but it works!

  DIDYOU KNOW?

  An electrical engineer uses integration to determine the exact length of electrical wire needed to connect two substations that are miles apart. If the wire were in straight lines, basic arithmetic could be used. But because the cable is hung from poles, it is constantly curving in the shape of a catenary. The use of calculus allows a very precise figure to be calculated.

  Area and the Definite Integral

  Overview: The second part of a typical calculus course is called integral calculus. Differential calculus concerned itself with the problem of finding the slope of a tangent line to a curve at a point. Armed with that information, using linear approximation, we could approximate a value of f (c) at point P close to the point of tangency.

  In integral calculus we are concerned with finding the area between a curve, the x-axis and lines x = a and x = b. Just as the slope of the tangent line had a mathematical name, the derivative, the area between a function f (x) and the x-axis between a and b is called a definite integral and is written as . This is different from an indefinite integral ∫ f(x) dx as studied in the previous section. The connection between the indefinite and definite integral will be studied later in this chapter when we look at the Fundamental Theorem of Calculus.

  The notation makes sense by looking at the left-hand figure below. We draw a rectangle whose height is f (x), the height of the rectangle above the x-axis. We define dx as the width of the rectangle. So f (x) dx represents (height) • (width) = the area of any one rectangle. The definite integral sign represents the sum of the areas of infinitely thin rectangles as shown in the right-hand figure below. The a represents the starting place for these rectangles while the b represents the ending place for these rectangles. Remember that definite integrals do not have a + C when calculated because the area is a specific number. That is why they are called definite

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  eBook - ePub

  AP® Calculus AB & BC Crash Course Book + Online

  • J. Rosebush, Stu Schwartz(Authors)
  • 2016(Publication Date)
  • Research & Education Association

    (Publisher)

  PART IV

  INTEGRALS

  Passage contains an image Chapter 12

  Types of Integrals, Interpretations and Properties of Definite Integrals, Theorems

  I.	TYPES OF INTEGRALS

  A.Indefinite integrals have no limits, ∫f(x)dx. This represents the antiderivative of f(x). That is, if ∫f(x)dx = F(x) + C, then F′(x) = f(x). When taking an antiderivative of a function, don’t forget to add C! For instance, ∫2xdx = x2 + C (The constant C is necessary because the antiderivative of f(x) = 2x could be F(x) = x2 or F(x) = x2 + 1 or F(x) = x2 – 2, and so on.) Sometimes, you are given an initial condition that allows you to find the value of C. For instance, find the antiderivative, F(x), of f(x) = 2x, given that F(0) = 1. Then, F(x) = ∫2xdx = x2 + C → F(0) = (0)2 + C = 1 → C = 1 → F(x) = x2 + 1. Another way of posing this question is: Find y if and y|x =0 = 1. The equation is called a differential equation (more on this later) because it contains a derivative.

  B.Definite integrals have limits x = a and . If f(x) is continuous on [a, b] and F′(x) = f(x), then (The First Fundamental Theorem of Calculus.)

  1.A definite integral value could be positive, negative, zero or infinity. When used to find area, the definite integral must have a positive value.

  i.	If f(x) > 0 on [a, b], then and geometrically it represents the area between the graph of f(x) and the x-axis on the interval [a, b]. For example, square units. Note that this could also have been solved geometrically because the area in question is that of a right triangle with a base of 3 units and a height of 6 units.
  Solving an area problem geometrically is really helpful when the question involves the integral of a piecewise linear function, for instance, . This represents the area between the function f(x) = |x| and the x-axis between x = –1 and x = 3. Noticing that this area is that of two right triangles, we have: units2

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