Question

The iterated integrals use spherical coordinates. Describe the solids determined by the limits of integration.

${\int }_{\frac{\pi }{2}}^{\pi }{\int }_0^{2\pi }{\int }_0^{2}f\left(\rho ,\theta ,\varphi \right){\rho }^2\sin \varphi d\rho d\theta d\varphi$

Short answer

The region represents sphere with second quadrant with the hemisphere below the $xy-$plane of the sphere with radius 2 centered at origin

Step-by-step solution

Step 1:Given information

The given expression is${\int }_{\frac{\pi }{2}}^{\pi }{\int }_0^{2\pi }{\int }_0^{2}f\left(\rho ,\theta ,\varphi \right){\rho }^2\sin \varphi d\rho d\theta d\varphi$

Step 2:Simplification

According to the order of integration

$\rho =0to\rho =2$

width="141">⇒x2+y2+z2=2

$\Rightarrow x^2+y^2+z^2=2^2$

This is a sphere equation

$\Rightarrow \theta$varies from $\theta =0to\theta =2\pi$

It means it is in the second quadrant

localid="1652082534348" $\Rightarrow \varphi$varies from$\frac{\mathrm{\pi }}{2}to\mathrm{\pi }$these the region represents the hemisphere below the xy -plane of the sphere with radius 2 centered at the origin