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Math Help - volume of solid using double integrals

  1. #1
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    volume 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
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  2. #2
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    Quote Originally Posted by tennisplaya999 View Post

    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
    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 z=\sqrt{r^2 - y^2} and you need to integrate z over

    R: \ x^2+y^2 \leq r^2, \ x \geq 0, \ y \geq 0. so choosing a suitable order of integration we'll have: 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}.
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