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Chapter 12: Problem 10

Use cylindrical coordinates to find the volume of the solid. Solid inside \(x^{2}+y^{2}+z^{2}=16\) and outside \(z=\sqrt{x^{2}+y^{2}}\)

Short Answer

Expert verified
The volume of the solid is \(V= \frac{32}{3}\pi\)

Step by step solution

01

Transform into cylindrical coordinates

Transform the equations of the sphere and cone into cylindrical coordinates. For the sphere: \(r^2+z^2=16\). For the cone: \(z=r\).
02

Define the limits of integration

Determine the limits of integration. Here, \(r\) goes from: \(0 \leq r \leq 4\), \(\theta\) goes from: \(0 \leq \theta \leq 2\pi\), and \(z\) goes from: \(r \leq z \leq \sqrt{16 - r^2}\).
03

Set up the integral

Set up the integral for volume using cylindrical coordinates. The volume \(V\) can be expressed as: \(V = \int_0^{2\pi} \int_0^4 \int_r^{\sqrt{16-r^2}} r\,dz\,dr\,d\theta\).
04

Calculate the integral

Calculate the integral. Doing so, the volume \(V\) is found to be: \(V= \frac{32}{3}\pi\).

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Most popular questions from this chapter

Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density or densities. (Hint: Some of the integrals are simpler in polar coordinates.) $$ y=9-x^{2}, y=0, \rho=k y^{2} $$

Mass In Exercises 13 and 14, use cylindrical coordinates to find the mass of the solid \(Q\). $$ \begin{array}{l} Q=\left\\{(x, y, z): 0 \leq z \leq 9-x-2 y, x^{2}+y^{2} \leq 4\right\\} \\ \rho(x, y, z)=k \sqrt{x^{2}+y^{2}} \end{array} $$

In Exercises 23-26, evaluate the improper iterated integral. $$ \int_{1}^{\infty} \int_{1}^{\infty} \frac{1}{x y} d x d y $$

In Exercises \(1-10\), evaluate the integral. $$ \int_{0}^{\cos y} y d x $$

In Exercises \(59-62,\) use a computer algebra system to approximate the iterated integral. $$ \int_{0}^{2 \pi} \int_{0}^{1+\cos \theta} 6 r^{2} \cos \theta d r d \theta $$
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