Find the volume of the solid of revolution generated by rotating the area bounded by y=x(2−x) and y=x about the y−axis.


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Question Number 214509 by depressiveshrek last updated on 10/Dec/24

$Find the volume of the solid of revolution$ $generated by rotating the area bounded$ $by y = x (2 - x) and y = x about the y - axis .$
Commented by mr W last updated on 11/Dec/24

$y o u s e e m n e v e r t o g i v e a n y f e e d b a c k$ $w h e n p e o p l e h a v e a n s w e r e d y o u r$ $q u e s t i o n s ?$
Commented by depressiveshrek last updated on 11/Dec/24

$Not true, sometimes I ask for more details$ $or disagree with the solution / proof . But$ $why does that matter ?$
Commented by mr W last updated on 11/Dec/24

$a c t u a l l y i t {d o e s n}^{'} t m a t t e r . i j u s t$ $t h o u g h t, n o r m a l l y w h e n s o m e b o d y$ $h a s r e c i e v e d t h e h e l p h e h a d w i s h e d,$ ${h e}^{'} l l s h o w s o m e r e a c t i o n, n o r m a l l y .$
Commented by depressiveshrek last updated on 11/Dec/24

$Okay then I will start to give feedback$ $from now on more frequently .$
	Answered by mr W last updated on 11/Dec/24

	$x = y$ $x^{2} - 2 x + 1 = 1 - y \Rightarrow x = 1 - \sqrt{1 - y}$ $1 - \sqrt{1 - y} = y \Rightarrow y = 0, 1$ $V = \int_{0}^{1} π [y^{2} - {(1 - \sqrt{1 - y})}^{2}] d y$ $= \int_{0}^{1} π (y^{2} + y - 2 + 2 \sqrt{1 - y}) d y$ $= π {[\frac{y^{3}}{3} + \frac{y^{2}}{2} - 2 y - \frac{4}{3} {(1 - y)}^{\frac{3}{2}}]}_{0}^{1}$ $= π (\frac{1}{3} + \frac{1}{2} - 2 + \frac{4}{3})$ $= \frac{π}{6} ✓$
	Commented by depressiveshrek last updated on 11/Dec/24

	$Thank you$
	Commented by mr W last updated on 11/Dec/24

	$a l t e r n a t i v e m e t h o d w i t h o u t$ $i n t e g r a l c a l c u l a t i o n :$ $V_{1} = \frac{2 \times 1 \times 1}{3} \times 2 π \times (\frac{5 \times 1}{8}) = \frac{5 π}{6}$ $V_{2} = \frac{1 \times 1}{2} \times 2 π \times (\frac{2 \times 1}{3}) = \frac{2 π}{3}$ $V = V_{1} - V_{2} = \frac{π}{6}$
	Commented by mr W last updated on 11/Dec/24

	Commented by mr W last updated on 11/Dec/24

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