Triple Integrals in Cylindrical and Spherical Coordinates

In each of the following questions, set up an iterated integral expression whose value determines the desired result. Then, evaluate the integral first by hand, and then using appropriate technology.

1. Find the volume of the “cap” cut from the solid sphere $x^2 + y^2 + z^2 = 4$ by the plane $z=1$, as well as the $z$-coordinate of its centroid.

2. Find the $x$-coordinate of the center of mass of the portion of the unit sphere that l…

For each of the following questions,

• sketch the region of integration,

• change the coordinate system in which the iterated integral is written to one of the remaining two,

• evaluate the iterated integral you deem easiest to evaluate by hand.

1. $\int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{\sqrt{x^2 + y^2}}^{\sqrt{2-x^2-y^2}} xy\, dz \, dy \, dx$

2. $\int_{0}^{\pi/2} \int_{0}^{\pi} \int_{0}^{1} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta$

3. $\int_{0}^{2\pi} \int_{0}^{1} \int_{r}^{1} r^2 \cos(\theta) \, dz \, dr \, d\theta$

Consider the solid region $S$ bounded above by the paraboloid $z = 16 - x^2 - y^2$ and below by the paraboloid $z = 3x^2 + 3y^2$.

1. Describe parametrically the curve in $\R^3$ in which these two surfaces intersect.

2. In terms of $x$ and $y$, write an equation to describe the projection of the curve onto the $xy$-plane.

3. What coordinate system do you think is most natural for an iterated integral that gives the volume of the solid?

4. Set up, but do not eval…