# Consider the region D in the first octant that is inside both of the sphere x^2+y^2 + z^2 =...

## Question:

Consider the region {eq}D {/eq} in the first octant that is inside both of the sphere {eq}x^2+y^2 + z^2 = 2^2 {/eq} and the cylinder {eq}x^2+y^2 =1 {/eq}. Set up a triple integral to find the volume of the solid in

a. rectangular coordinates

b. cylindrical coordinates

c. spherical coordinates

## Calculating Volume using Rectangular, Cylindrical, and Spherical Coordinates:

If we wish to find the volume of a region {eq}D {/eq} in three dimensions, we can use a triple integral. Depending on the coordinate system we use, we set up the integrals with different limits of integration. The volume integral is {eq}V = \displaystyle\iiint_D dV. {/eq}

In rectangular coordinates we use the differential {eq}dV = dz \: dy \: dx. {/eq}

In cylindrical coordinates we use the differential {eq}dV = r \: dz \: dr \: d\theta. {/eq}

In spherical coordinates we use the differential {eq}dV = \rho^2 \sin \phi \: d\rho \: d\phi \: d\theta. {/eq}