hi again :) i need help setting this up using double integrals

Find the volume of the given soild: Bounded by the cylinders x^2+y^2=r^2 and y^2+z^2=r^2.

thanks

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- Apr 19th 2009, 12:18 PMtennisplaya999volume of solid using double integrals
hi again :) i need help setting this up using double integrals

Find the volume of the given soild: Bounded by the cylinders x^2+y^2=r^2 and y^2+z^2=r^2.

thanks - Apr 19th 2009, 04:00 PMNonCommAlg
you only need to find the volume of the solid in the first octant and then multiply the result by 8 to get the full volume. so you have $\displaystyle z=\sqrt{r^2 - y^2}$ and you need to integrate $\displaystyle z$ over

$\displaystyle R: \ x^2+y^2 \leq r^2, \ x \geq 0, \ y \geq 0.$ so choosing a suitable order of integration we'll have: $\displaystyle V=8 \int_0^r \int_0^ {\sqrt{r^2 - y^2}} \sqrt{r^2 - y^2} \ dx \ dy=8\int_0^r(r^2 - y^2) \ dy = \frac{16r^3}{3}.$