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Find the volume V of the solid obtained by rotating the regionbounded by the given...

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x2, x = y2; about y = 1

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Answer #1

We have y= x^2,y= \sqrt x

Now we find the points of intersection first:

x^2= \sqrt x

\Rightarrow x= 0,1

So when x is 0 y is also 0

and when x is 1, y is also 1

Hence the points of intersection are (0,0) and (1,1)

Given it is rotated about y= 1

So using disk method, the volume is given by:

V= \pi \int_{0}^{1}\left ( (x^2-1)^2-(\sqrt x -1)^2 \right )dx

V= \pi \int_{0}^{1}\left ( x^4-2x^2+1-x+2\sqrt x -1 \right )dx

V= \pi \int_{0}^{1}\left ( x^4-2x^2-x+2\sqrt x \right )dx

V= \pi \left [\frac{x^5}{5}-\frac{2x^3}{3}-\frac{x^2}{2}+\frac{4}{3} x^{\frac{3}{2}} \right ]_{0}^{1}

V= \pi \left [\frac{1}{5}-\frac{2}{3}-\frac{1}{2}+\frac{4}{3} \right ]

V= \frac{11}{30}\pi

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