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Chapter 4: Q. 27 (page 353)

Use geometry (i.e., areas of triangles, rectangles, and circles) to find the exact values of each of the definite integrals in Exercises 21-28.

鈭�03(1-|x-1|)dx.

Short Answer

Expert verified

The exact value of鈭�03(1-|x-1|)dxis,12.

Step by step solution

01

Step 1. Given information

鈭�03(1-|x-1|)dx.

02

Step 2. The following figure shows the graph of (1-|x-1|).

03

Step 3. Find the area.

The area of the shaded triangle is,

12脳1脳2-12脳2脳12=1-12=12

Therefore, the exact value is,12.

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

鈭�3x2-4dx

Use the graph of f to estimate the values of A(1), A(2), A(3)

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

鈭�蟺/4蟺/2鈥�csc鈦�xcot鈦�xdx

Given a simple proof that if n is a positive integer and c is any real number, then鈭�k=1nc=cn

Show by exhibiting a counterexample that, in general, 鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx. In other words, find two functions f and g such that the integral of their quotient is not equal to the quotient of their integrals.

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