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Calculus Level 4

Determine the volume of the solid obtained by rotating the region bounded by y = 2 x − 1 y=2\sqrt{x-1} and y = x − 1 y=x-1 about the line x = − 1 x=-1 .

If your answer comes in form of p π q \dfrac{p \pi} {q} , where p p and q q are co- prime integers, find the value of p − q p-q .


The answer is 91.

Anish Harsha by Anish Harsha , 20, India

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1 solution

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Apr 20, 2016

Solve each for y and subtract -1 to get the correct radius. x = y + 2 , x = y 2 4 + 2 x = y+2, x = \frac{y^2}{4}+2 . The functions meet at y = 4 and y = 0 so these are the limits for the integral. Using area of a circle = π r 2 \pi r^2 and subtracting the inner function from the outer function.

V = π ∫ 0 4 ( y + 2 ) 2 d y − ( y 2 4 + 2 ) 2 d y = 96 π 5 V = \displaystyle\pi\int_{0}^{4}(y+2)^2dy - ( \frac{y^2}{4}+2)^2dy = \boxed{\frac{96\pi}{5}}

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