Find the volume of the solid

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December 2nd 2006, 05:33 AM
Yogi_Bear_79

Find the volume of the solid

Find the volume of the solid that is generated by rotating the region formed by the graphs of y = x2, y = 2, and x = 0 about the y-axis.
December 2nd 2006, 05:55 AM
galactus

You can use shells or washers.

Shells:
$2{\pi}\int_{0}^{\sqrt{2}}x(2-x^{2})dx$

Washers:
${\pi}\int_{0}^{2}ydy$
December 2nd 2006, 10:55 AM
Yogi_Bear_79

sorry to be so ignorant at this, but using shells could you show me the rest of the answer?
December 2nd 2006, 11:45 AM
Jameson

$2{\pi}\int_{0}^{\sqrt{2}}x(2-x^{2})dx$

$2 \pi \int_{0}^{\sqrt{2}} 2x-x^3dx$

Call the integral $F(x)=x^2-\frac{x^3}{3}$

Thus the volume is $2\pi \left( F(\sqrt{2})-F(0) \right)$

Make sense?
December 2nd 2006, 11:55 AM
Yogi_Bear_79

so if I did it right then, V = 6.64257 ?

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