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Question

Sketch the region bounded by the curves y=x2 and y=21+x2. Find the area.

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Solution

The given curves are y=x2
and y=21+x2
Solving (1) and (2), we have
x2=21+x2
x2+x22=0
(x21)(x2+2)=0
x=±1
Also, y=21+x2 is an even function.
Hence, its graph is symmetrical about y-axis.
At x=0,y=2, by increasing the values of x,y decreases and when x,y0.
y=0 is an asymptote of the given curve.
Thus, the graph of the two curves is as follows
Ref. image
The required area
=210(21+x2x2)dx
=(4tan1x2x33)10
=π23sq. units
1973578_1786717_ans_bbecbd8e17d04ccda16e14d5ccf79e54.jpg

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