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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Find the area of the region bounded by the given curves.

y = 1 2 x 2 ,    y = | x |

To determine

To find:

The area of the region bounded by the given curves.

Explanation

1) Concept:

The area A of the region bounded by the curves y=fx&  y=gx and the lines x=a, x=b, where f and g are continuous and fxg(x) for all x in a, b, is

A= abfx-gxdx

2)  Given:

The region bounded by curves y=1-2x2 and y=|x|.

3) Calculation:

As the given region is bounded by curves  y=1-2x2 and y=|x|,

From the graph, the region is symmetric about y axis.Therefore, consider x0 double the given area.

If x0, then x=x, and the graphs intersect when x=1-2x2. That is when

2x2+x-1=0

2x-1x+1=0

Therefore, either x=12 or x=-1, but -1<0 so x=12

Hence, the limits of integration is from x=0 to x=12

The upper boundary curve is y=1-2x2 and the lower boundary curve is y=x

Therefore, the area of the region bounded

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