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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Sketch the region enclosed by the given curves and find its area.

x = 2 y 2 ,   x = 4 + y 2

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 x=f(y), x=g(y) and the lines y=c and y=d  is

A= cdfy-gydy

Let’s denote the right boundary curve by xR and the left boundary by xL, then the area between these curves is given by

A=cdxR-xLdy

2) Given:

x=2y2 and   x=4+y2

3) Calculation:

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

2y2=4+y2

y2=4

Thus, the points of intersection are at y=-2 and   y=2. The region is sketched in the following figure.

Here, the right curve is x=4+y2 and the left curve is x=2y2. Therefore,

xR=4+y2

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