what-is-the-volume-of-region-bounded-by-y-x-2-2x-and-y-x-that-is-rotated-about-y-4-

Question Number 102606 by bemath last updated on 10/Jul/20

w h a t i s t h e v o l u m e o f r e g i o n

b o u n d e d b y y = x^{2} - 2 x a n d

y = x t h a t i s r o t a t e d a b o u t

y = 4 ?

Commented by bemath last updated on 10/Jul/20

Answered by Ar Brandon last updated on 10/Jul/20

Volume, V = π \int_{0}^{3} {{(x^{2} - 2 x - 4)}^{2} - {(x - 4)}^{2}} dx

Answered by bobhans last updated on 10/Jul/20

v o l = π {\int^{3}}_{0} (4^{2} - {(x^{2} - 2 x)}^{2}) - (4^{2} - x^{2}) d x

= π \underset{0}{\int^{3}} (x^{2} - {(x^{2} - 2 x)}^{2} d x

= π \underset{0}{\int^{3}} (x^{2} + x^{2} - 2 x) (2 x) d x

= 4 π \underset{0}{\int^{3}} x (x^{2} - x) d x = 4 π \underset{0}{\int^{3}} (x^{3} - x^{2}) d x

= 4 π {\frac{1}{4} x^{4} - \frac{1}{3} x^{3}}_{0}^{3} = 4 π {\frac{81}{4} - \frac{27}{3}}

= 4 π {\frac{45}{4}} = 45 π

Commented by Ar Brandon last updated on 10/Jul/20

What difference will it make if it was rather rotated about y=0 with respect to your solution ? ��

Commented by bemath last updated on 10/Jul/20

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