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Question Number 74231 by necxxx last updated on 20/Nov/19

Commented by necxxx last updated on 20/Nov/19

Commented by necxxx last updated on 20/Nov/19

$h e r e w a s a s t e p t a k e n t h o u g h n o t s o$ $s u r e . P l e a s e I n e e d h e l p . T h a n k s i n$ $a d v a n c e .$
	Answered by mind is power last updated on 20/Nov/19

	$Z = \sqrt{x^{2} + y^{2}} \cap (x^{2} + y^{2} + z^{2} = 16) \Rightarrow 2 (x^{2} + y^{2}) = 16$ $\Rightarrow x^{2} + y^{2} = 8$ $i n t e r s e c t i o n o v e r c i r c l e c e n t e r$ $0 ⩽ r ⩽ 4, - \frac{π}{2} ⩽ θ ⩽ \frac{π}{2}, - π ⩽ α ⩽ π$ $x = r c o s (θ) c o s (α)$ $y = r c o s (θ) s i n (α)$ $z = r s i n (θ)$ $0 ⩽ Z ⩽ \sqrt{x^{2} + y^{2}} \Rightarrow$ $0 ⩽ r s i n (θ) ⩽ \sqrt{r^{2} {c o s}^{2} (θ)} = r c o s (θ)$ $\Rightarrow 0 ⩽ s i n (θ) ⩽ c o s (θ) \Rightarrow θ \in [0, \frac{π}{4}]$ $V = \int_{0}^{4} \int_{0}^{\frac{π}{4}} \int_{- π}^{π} r^{2} s i n (θ) d α d θ d r$ $= \int_{0}^{4} r^{2} d r . \int_{0}^{\frac{π}{4}} s i n (θ) d θ . \int_{- π}^{π} d α$ $= \frac{64}{3} . (1 - \frac{\sqrt{2}}{2}) . 2 π$ $= (2 - \sqrt{2}) . \frac{64 π}{3}$
	Commented by necxxx last updated on 25/Nov/19

	$o h . . . T h a n k y o u s o m u c h .$
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