   Chapter 8.2, Problem 98E

Chapter
Section
Textbook Problem

Finding a Pattern Find the area bounded by the graphs of y = x sin   x   and  y = 0 over each interval.(a) [ 0 ,   π ] (b) [ π ,   2 π ] (c) [ 2 π ,   3 π ] Describe any patterns that you notice. What is the area between the graphs of y = x sin   x   and  y = 0 over the interval [ n π ,   ( n + 1 ) π ] , where n is any nonnegative integer? Explain.

(a)

To determine

To calculate: The area bounded by the graphs of y=xsinx and y=0 over the interval [0,π].

Explanation

Given:

The functions are:y=xsinx and y=0.

Formula used:

The area bounded by the two curves f(x) and g(x) over the interval (a,b) is given as:

A=ab[f(x)g(x)]dx.

Integration by parts:

If u and v are the function of x then, udv=uvvdu

Calculation:

Consider the area bounded by the graphs of y=xsinx and y=0 is:

A=0πxsinxdx

To apply integration by parts,

Let u=x then du=1

and

dv=sinxdxv=sinxdx=&#

(b)

To determine

To calculate: The area bounded by the graphs of y=xsinx and y=0 over the interval [π,2π].

(c)

To determine

To calculate: The area bounded by the graphs of y=xsinx and y=0 over the interval [2π,3π].

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