Find-the-surface-area-of-a-solid-that-is-common-part-of-two-cylinders-x-2-y-2-a-2-y-2-z-2-a-2-Compute-the-volume-also-

Question Number 20471 by ajfour last updated on 27/Aug/17

F i n d t h e s u r f a c e a r e a o f a s o l i d

t h a t i s c o m m o n p a r t o f t w o

c y l i n d e r s x^{2} + y^{2} = a^{2}, y^{2} + z^{2} = a^{2} .

C o m p u t e t h e v o l u m e a l s o .

Commented by ajfour last updated on 27/Aug/17

Commented by ajfour last updated on 27/Aug/17

S = 16 \int_{0}^{π / 2} x a d θ = 16 a^{2} \int_{0}^{π / 2} \cos θ d θ

= 16 a^{2} . (s i n c e x = z = a \cos θ) .

Answered by ajfour last updated on 28/Aug/17

V = 16 \int_{0}^{a} \int_{0}^{a \cos θ} (a \cos θ - x) d x] d y

= 16 \int_{0}^{π / 2} (a x \cos θ - \frac{x^{2}}{2}) ∣_{0}^{a \cos θ} (a \cos θ) d θ

= 16 \int_{0}^{π / 2} \frac{a^{2} \cos^{2} θ}{2} (a \cos θ) d θ

= 8 a^{3} \int_{0}^{π / 2} (1 - \sin^{2} θ) d (\sin θ)

= 8 a^{3} [\sin θ - \frac{\sin^{3} θ}{3}] ∣_{0}^{π / 2}

V = \frac{16 a^{3}}{3} .

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