   Chapter 5.1, Problem 19E

Chapter
Section
Textbook Problem

# Sketch the region enclosed by the given curves and find its area. y = cos π x ,   y = 4 x 2 − 1

To determine

To:

Sketch the region and find the enclosed area.

Explanation

1) Concept:

Formula:

The area A of the region bounded by the curves   y=f(x), y=g(x) and the lines x=a and x=b  is

A= abfx-gxdx

fx-gx=fx-gx when fxg(x)gx-fx when gxf(x)

2) Given:

y=cosπx and   y=4x2-1

3) Calculation:

The point of intersection occurs when both equations are equal to each other, that is,

cosπx=4x2-1

As  -1cosθ1

So at point of intersection,

cosπx= 4x2-1

That is  4x2-cosπx-1=0

By quadratic formula thus x=±0-4(4)(-cosπx-1)2(4)=±16(cosπx+1)8=±(cosπx+1)2 Thus by trial and error (in view of the graph) points of intersection are at x=-12 and   x=12. The region is sketched in the following figure. We may also directly use the graph generated by some graphing utility to find the points of intersections.

Here,cosπx4x2-1 when  -12x12

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