find the volume of the solid generated by thr revolution of the curve y(x^2 +a^2 )=a^3 about itd asymptote.


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Question Number 25745 by rita1608 last updated on 13/Dec/17

$f i n d t h e v o l u m e o f t h e s o l i d$ $g e n e r a t e d b y t h r r e v o l u t i o n o f t h e$ $c u r v e y (x^{2} + a^{2}) = a^{3} a b o u t i t d$ $a s y m p t o t e .$
	Answered by jota@ last updated on 15/Dec/17

	$a s i n t o t a y = 0$ $V = 2 \int_{0}^{\infty} π y^{2} d x$ $V = 2 π a^{6} \int_{0}^{\infty} \frac{d x}{{(x^{2} + a^{2})}^{2}} .$ $x = a t a n z d x = {a s e c}^{2} z d z$ $x^{2} + a^{2} = a^{2} ({t a n}^{2} z + 1) = a^{2} {s e c}^{2} z$ $V = 2 π a^{6} \int_{0}^{π / 2} \frac{{c o s}^{2} z d z}{a^{3}}$ $= 2 π a^{3} \int_{0}^{π / 2} \frac{1 + c o s 2 z}{2} d z$ $= π a^{3} (z + \frac{s i n 2 z}{2}) ∣_{0}^{π / 2}$ $= \frac{π^{2} a^{3}}{2} .$
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