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Chapter 13: Problem 24

Find the mass of the following objects with the given density functions. The solid cylinder \(D=\\{(r, \theta, z): 0 \leq r \leq 3,0 \leq z \leq 2\\}\) with density \(\rho(r, \theta, z)=5 e^{-r}\)

Short Answer

Expert verified
Question: Calculate the mass of a solid cylinder with radius 3 and height 2, given the density function, 蟻(r, 胃, z) = 5e^(-r). Answer: The mass of the solid cylinder is 20蟺 - 80蟺 e^(-3).

Step by step solution

01

Identify the limits of integration for r, 胃, and z

The given solid cylinder has the ranges of coordinates \(0 \leq r \leq 3, 0 \leq z \leq 2\), and since it is a full cylinder, the angle \(\theta\) ranges from \(0\) to \(2\pi\). So our limits for integration are as follows: - For \(r\): \(0\) to \(3\) - For \(\theta\): \(0\) to \(2\pi\) - For \(z\): \(0\) to \(2\)
02

Set up the triple integral

Now, we will set up the triple integral using the given density function \(\rho(r, \theta, z) = 5e^{-r}\) and the limits of integration we found in step 1: $$M = \int_{0}^{2} \int_{0}^{2\pi} \int_{0}^{3} 5e^{-r} r \, dr \, d\theta \, dz$$
03

Perform the r-integration

First, we will integrate with respect to r: $$M = \int_{0}^{2} \int_{0}^{2\pi} \left[-5e^{-r}(r+1)\right]_0^3 \, d\theta \, dz$$
04

Simplify the integrand

Let's simplify the expression inside bracket by substituting the limits: $$M = \int_{0}^{2} \int_{0}^{2\pi} \left[-5e^{-3}(3+1) + 5e^{0}(0+1)\right] \, d\theta \, dz$$ $$M = \int_{0}^{2} \int_{0}^{2\pi} \left(5-20e^{-3}\right) \, d\theta \, dz$$
05

Perform the 胃-integration

Now, integrate with respect to \(\theta\): $$M = \int_{0}^{2} \left[(5-20e^{-3})\theta\right]_{0}^{2\pi} \, dz$$ $$M = \int_{0}^{2} (10\pi-40\pi e^{-3}) \, dz$$
06

Perform the z-integration

Finally, integrate with respect to z: $$M = \left[(10\pi-40\pi e^{-3})z\right]_{0}^{2}$$
07

Calculate the mass of the solid cylinder

Compute the mass by substituting the limit: $$M = (20\pi-80\pi e^{-3})$$ So the mass of the solid cylinder is \(20\pi-80\pi e^{-3}\).

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Most popular questions from this chapter

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