Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Integral chapter 6 The Indefinite Integral Substitution The Definite Integral. - презентация

1 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Integral chapter 6 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental Theorem of Calculus

2 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Antiderivative An antiderivative of a function f is a function F such that Ex.An antiderivative of since is

3 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. means to find the set of all antiderivatives of f. The expression: read the indefinite integral of f with respect to x, Integral sign Integrand Indefinite Integral x is called the variable of integration

4 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Notice Constant of Integration Represents every possible antiderivative of 6x.

5 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Power Rule for the Indefinite Integral, Part I Ex.

6 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Power Rule for the Indefinite Integral, Part II Indefinite Integral of e x and b x

7 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sum and Difference Rules Ex. Constant Multiple Rule Ex.

8 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Integral Example/Different Variable Ex. Find the indefinite integral:

9 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Position, Velocity, and Acceleration Derivative Form If s = s(t) is the position function of an object at time t, then Velocity = v =Acceleration = a = Integral Form

10 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Integration by Substitution Method of integration related to chain rule differentiation. If u is a function of x, then we can use the formula

11 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Integration by Substitution Ex. Consider the integral: Sub to getIntegrateBack Substitute

12 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Evaluate Pick u, compute du Sub in Integrate

13 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Evaluate

14 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Evaluate

15 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Shortcuts: Integrals of Expressions Involving ax + b Rule

16 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Riemann Sum If f is a continuous function, then the left Riemann sum with n equal subdivisions for f over the interval [a, b] is defined to be

17 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Definite Integral If f is a continuous function, the definite integral of f from a to b is defined to be The function f is called the integrand, the numbers a and b are called the limits of integration, and the variable x is called the variable of integration.

18 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Approximating the Definite Integral Ex. Calculate the Riemann sum for the integral using n = 10.

19 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Definite Integral is read the integral, from a to b of f(x)dx. Also note that the variable x is a dummy variable.

20 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Definite Integral As a Total If r(x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

21 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Definite Integral As a Total Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

22 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Area Under a Graph a b Idea: To find the exact area under the graph of a function. Method: Use an infinite number of rectangles of equal width and compute their area with a limit. Width: (n rect.)

23 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Approximating Area Approximate the area under the graph of using n = 4.

24 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Area Under a Graph a b f continuous, nonnegative on [a, b]. The area is

25 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Geometric Interpretation (All Functions) Area of R 1 – Area of R 2 + Area of R 3 a b R1R1 R2R2 R3R3

26 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Area Using Geometry Ex. Use geometry to compute the integral Area = 2 Area =4

27 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Fundamental Theorem of Calculus Let f be a continuous function on [a, b]. 2. If F is any continuous antiderivative of f and is defined on [a, b], then

28 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Fundamental Theorem of Calculus Ex.

29 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Evaluating the Definite Integral Ex. Calculate

30 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Substitution for Definite Integrals Ex. Calculate Notice limits change

31 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Computing Area Ex. Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of Gives the area since 2x 3 is nonnegative on [0, 2]. AntiderivativeFund. Thm. of Calculus