Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Integration 5 Antiderivatives Substitution Area Definite Integrals Applications

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Antiderivative A function F is an antiderivative of f on an interval I if for all x in I. Ex.is an antiderivative of Since

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Let G be an antiderivative of f. Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Theorem Notice are all antiderivatives of

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. The process of finding all antiderivatives of a function is called integration. means to find all the antiderivatives of f. The Notation:read “the integral of f,” Integral signIntegrand Indefinite integral

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Basic Rules RuleExample

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Basic Rules Rule Ex. Find the indefinite integral

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Initial Value Problem To find a function F that satisfies the differential equation and one or more initial conditions. Ex. Find a function f if it is known that: Gives C = 3

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Integration by Substitution Method of integration related to chain rule differentiation. Ex. Consider the integral: Sub to getIntegrateBack Substitute

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Integration by Substitution Steps: 1. Pick u = f (x), often the “inside function.” 2.Compute 4. Substitute to express the integral in terms of u. 5. Integrate the resulting integral. 6. Substitute to get the answer in terms of x. 3.Solve for and replace dx

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Evaluate Pick u, compute du Sub in Integrate

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Evaluate

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Width: (n rect.) Approximating the Area Under the Graph of a Function Use rectangles, each with equal width. Three techniques: Left end pointsRight end pointsMidpoints

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Example Approximate the area under the graph of using n = 4; midpoints midpoints

Copyright (c) 2003 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.)

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Area Under a Graph a b f continuous, nonnegative on [a, b]. The area, A is x n, x n, …,x n are arbitrary, n subintervals each with width (b - a)/n.

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. 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. 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) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Integrability of a Function Let f be continuous on [a, b]. Then f is integrable on [a, b]; that is, exists. Geometric Interpretation Area of R 1 – Area of R 2 + Area of R 3 a b R1R1 R2R2 R3R3

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

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Fundamental Theorem of Calculus Let f be continuous on [a, b]. Then where F is any antiderivative of f.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Finding the Area Under a Curve Ex. Find the area under the graph of Gives the area since 2x 3 is nonnegative on [0, 2]. AntiderivativeFund. Thm. of Calculus

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Evaluating the Definite Integral Ex. Evaluate

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Properties of the Definite Integral (c is a constant)

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Average Value of a Function If f is integrable on [a, b], then the average value of f over [a, b] is Ex. Find the average value of

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Area Between Two Curves a b Let f and g be continuous functions, the area bounded above by y = f (x) and below by y = g(x) on [a, b] is provided that R

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Find the area bounded by the curves and the vertical lines x = – 1 and x = 2.