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Find the area bounded by the curves y=s...

Find the area bounded by the curves `y=sqrt(1-x^(2))` and `y=x^(3)-x` without using integration.

Text Solution

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We have `y=sqrt(1-x^(2)) " " (i) `
and `y=x^(3)-x=x(x-1)(x+1) " " (ii)`
Graph of (i) is the semicircle `x^(2)+y^(2)=1` , above the x-axis.
Equation (ii) is an odd function and the graph is symmetrical about (0, 0) , intersecting the x-axis at `x=0, +-1`.
The graphs of (i) and (ii) are as shown in the following figure.

Required area, A = Area of region BHOGACB
Now Area of region BEOHB = Area of region OGAFO
Hence A = Area of of semicircle
`=pi//2` sq. units
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