Use an appropriate coordinate system to find the volume of the solid lying along the positive z-axis and bounded by the cone and the sphere .

Question

Use an appropriate coordinate system to find the volume of the solid lying along the positive z-axis and bounded by the cone  and the sphere  .

Quizplus · Accepted Answer

We can use cylindrical coordinates since the solid has rotational symmetry around the z-axis.

The cone $z = \sqrt{x^2+y^2}$ can be expressed in cylindrical coordinates as $z=r$. The sphere $x^2+y^2+z^2=16$ becomes $r^2+z^2=16$.

To find the limits of integration, we note that the cone intersects the sphere when $r^2+z^2=16$ and $z=\sqrt{r^2+z^2}$, which gives $r^2 = \frac{16}{2+\sqrt{5}}$ and $z^2 = \frac{32}{2+\sqrt{5}}$. This means that we can integrate $r$ from 0 to $\frac{2\sqrt{2}}{\sqrt{2+\sqrt{5}}}$ and $	heta$ from 0 to $2\pi$. For each value of $r$ and $	heta$, the limits of $z$ are from $r$ to $\sqrt{16-r^2}$.

Thus, the volume integral is given by:

$$\int_0^{2\pi} \int_0^{\frac{2\sqrt{2}}{\sqrt{2+\sqrt{5}}}} \int_r^{\sqrt{16-r^2}} r dz dr d	heta$$

Evaluating this integral gives the answer B,  $\frac{64}{3(2+\sqrt{5})}\pi$.

Use an Appropriate Coordinate System to Find the Volume of the Solid