91Ó°ÊÓ

Evaluate the spherical coordinate integrals. \(\int_{0}^{3 \pi / 2} \int_{0}^{\pi} \int_{0}^{1} 5 \rho^{3} \sin ^{3} \phi d \rho d \phi d \theta\)

In Exercises \(21-30,\) sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$ \int_{0}^{1} \int_{1-x}^{1-x^{2}} d y d x $$

The region in the first octant bounded by the coordinate planes, the plane \(y+z=2,\) and the cylinder \(x=4-y^{2}\)

In Exercises \(21-30,\) sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$ \int_{0}^{1} \int_{1}^{e^{x}} d y d x $$

Mass of a plate Find the mass of a thin plate covering the region outside the circle \(r=3\) and inside the circle \(r=6 \sin \theta\) if the plate's density function is \(\delta(x, y)=1 / r\)

Evaluate the spherical coordinate integrals. \(\int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{\sec \phi}^{2} 3 \rho^{2} \sin \phi d \rho d \phi d \theta\)

A parabolic rain gauge \(A\) bowl is in the shape of the graph of \(z=x^{2}+y^{2}\) from \(z=0\) to \(z=10\) in. You plan to calibrate the bowl to make it into a rain gauge. What height in the bowl would correspond to 1 in. of \(\operatorname{rain} ? 3\) in. of rain?

Finding a centroid Find the centroid of the region cut from the first quadrant by the circle \(x^{2}+y^{2}=a^{2} .\)

Evaluate the spherical coordinate integrals. \(\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{\sec \phi}(\rho \cos \phi) \rho^{2} \sin \phi d \rho d \phi d \theta\)

In Exercises \(21-30,\) sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$ \int_{0}^{\ln 2} \int_{e^{x}}^{2} d x d y $$