Compute the volume of a solid bounded by a surface with equation (x^2 +y^2 +z^2 )^2 =a^3 x .


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Question Number 20162 by ajfour last updated on 23/Aug/17

$C o m p u t e t h e v o l u m e o f a s o l i d$ $b o u n d e d b y a s u r f a c e w i t h e q u a t i o n$ ${(x^{2} + y^{2} + z^{2})}^{2} = a^{3} x .$
	Answered by ajfour last updated on 23/Aug/17

	$s u r f a c e i s s y m m e t r i c a l a b o u t$ $x a x i s a n d r a n g e s f r o m x = 0 t o$ $x = a .$ $L e t x^{2} + y^{2} = r^{2}, T h e n$ $x^{2} + r^{2} = a \sqrt{a} (\sqrt{x})$ $V o l u m e V = \int_{0}^{a} π r^{2} d x$ $= π \int_{0}^{a} [a \sqrt{a} (\sqrt{x}) - x^{2}] d x$ $= π [\frac{2 a \sqrt{a} (x \sqrt{x})}{3} - \frac{x^{3}}{3}] ∣_{0}^{a}$ $V = \frac{π}{3} a^{3} .$
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