Download presentation

Published byElmer Cannon Modified over 5 years ago

1
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

The Integral chapter6 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental Theorem of Calculus Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 is since Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

3
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

4
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

5
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

6
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

7
**Sum and Difference Rules**

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

8
**Integral Example/Different Variable**

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

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

10
**Integration by Substitution**

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

11
**Integration by Substitution**

Ex. Consider the integral: Sub to get Integrate Back Substitute Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

12
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

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

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

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

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

15
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

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

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

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

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

20
**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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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

22
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

Area Under a Graph Width: (n rect.) 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. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

23
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

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

25
**Geometric Interpretation (All Functions)**

a b R2 Area of R1 – Area of R2 + Area of R3 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

26
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

27
**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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

28
**Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.**

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

29
**Evaluating the Definite Integral**

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

30
**Substitution for Definite Integrals**

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

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 2x3 is nonnegative on [0, 2]. Antiderivative Fund. Thm. of Calculus Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google