4.8 Triple Integrals in Cylindrical and Spherical Coordinates

Triple Integrals in Cylindrical and Spherical Coordinates

Introduction

Triple Integrals in Cylindrical Coordinates

Triple Integrals in Spherical Coordinates

In this Preview Activity, we will be reviewing the meaning of each of the cylindrical and spherical coordinates by looking at a description of a common surface in either cylindrical or spherical coordinates. For each task, you should draw a plot of the surface described by hand and write a few sentences describing how your plot relates to the cylindrical/spherical coordinates.

What familiar surface is described by the points in cylindrical coordinates with $r=2$, $0 \leq \theta \leq 2\pi$, and $0 \leq z \leq 2$? How does this example suggest that we call these coordinates cylindrical coordinates? How does the surface change if we restrict $\theta$ to $0 \leq \theta \leq \pi$?

Let $S$ be the solid bounded above by the graph of $z = x^2+y^2$ and below by $z=0$ on the unit disk in the $xy$-plane.

Suppose the density of the cone defined by $r = 1 - z$, with $z \geq 0$, is given by $\delta(r, \theta, z) = z$. A picture of the cone is shown in Figure. Set up an iterated integral in cylindrical coordinates that gives the mass of the cone. You do not need to evaluate this integral.

Write an iterated integral expression in cylindrical coordinates whose value is the volume of the solid bounded below by the cone $z = \sqrt{x^2+y^2}$ and above by the cone $z = 4 - \sqrt{x^2+y^2}$. A picture is shown in Figure. You do not need to evaluate this integral.

Recall that the sphere of radius $a$ has spherical equation $\rho = a$. Set up and evaluate an iterated integral in spherical coordinates to determine the volume inside a sphere of radius $a$.

Set up, but do not evaluate, an iterated integral expression in spherical coordinates whose value is the mass of the solid obtained by removing the region inside the cone $\phi=\frac{\pi}{4}$ from the sphere $\rho = 2$. Let $\delta$ at the point $(x,y,z)$ be the density given by $\delta(x,y,z) = \sqrt{x^2+y^2+z^2}$. An illustration of this solid is shown in Figure.