# Homework Homework Assignment #2 Read Section 5.3

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Homework Homework Assignment #2 Read Section 5.3
Page 321, Exercises: 1 – 61(EOO), 71, skip 41 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Draw a graph of the signed area represented by the integral and compute it using geometry. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Draw a graph of the signed area represented by the integral and compute it using geometry. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Draw a graph of the signed area represented by the integral and compute it using geometry. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 13. Evaluate the integrals for f (x) shown in Figure 14. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Sketch the signed area represented by the integral. Indicate the regions of positive and negative area. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Determine the sign of the integral without calculating it. Draw a graph if necessary. The sign of the integral is negative. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Calculate the Riemann sum for the given function, partition, and choice of intermediate points. Also, sketch the graph of f and the rectangles corresponding to R (f, P, C). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Use the basic properties of the integral and the formulas in the summary to calculate the integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Use the basic properties of the integral and the formulas in the summary to calculate the integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Use the formulas in the summary and Equation 9 to evaluate the integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Use the formulas in the summary and Equation 9 to evaluate the integral. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Calculate the integral, assuming Rogawski Calculus

Example, Page 321 Calculate the integral, assuming Rogawski Calculus

Example, Page 321 Express each integral as a single integral.

Example, Page 321 Calculate the integral, assuming f is an integrable function such . Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 321 Calculate the integral. Rogawski Calculus

Calculus, ET First Edition
Jon Rogawski Calculus, ET First Edition Chapter 5: The Integral Section 5.3: The Fundamental Theorem of Calculus, Part I Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

The Fundamental Theorem of Calculus ties together the operations of
differentiation and integration in a relationship. Specifically, the integral of f (x) evaluated from a to b is equal to the antiderivative of f (x) evaluated at x = b minus the antiderivative of f (x) evaluated at x = a. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Verify the assertion about the area under the curve in Figure 2.

Find the area under the curve between the limits shown in figure 3.

Verify that the shaded are in Figure 4 is 2. Rogawski Calculus

Calculate the shaded area in Figure 5.

Recalling that the antiderivative of f (x) = x–1 is F(x) = ln x, the FTC
tells us: Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Verify the relationship in the caption to Figure 6.

Find the areas of the shaded regions in Figure 7. Rogawski Calculus

Basic Antiderivative Formulae

Example, Page 329 Evaluate the integral using the FTC 1.

Example, Page 329 Evaluate the integral using the FTC 1.

Example, Page 329 Evaluate the integral using the FTC 1.

Example, Page 329 Evaluate the integral using the FTC 1.

Example, Page 329 Evaluate the integral using the FTC 1.

Example, Page 329 Write the integral as a sum of integrals without the absolute values and evaluate. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Example, Page 329 Evaluate the integral in terms of the constants.