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.

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- Dec 2nd 2006, 05:33 AMYogi_Bear_79Find 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.

- Dec 2nd 2006, 05:55 AMgalactus
You can use shells or washers.

Shells:

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

Washers:

$\displaystyle {\pi}\int_{0}^{2}ydy$ - Dec 2nd 2006, 10:55 AMYogi_Bear_79
sorry to be so ignorant at this, but using shells could you show me the rest of the answer?

- Dec 2nd 2006, 11:45 AMJameson
$\displaystyle 2{\pi}\int_{0}^{\sqrt{2}}x(2-x^{2})dx$

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

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

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

Make sense? - Dec 2nd 2006, 11:55 AMYogi_Bear_79
so if I did it right then, V = 6.64257 ?