Problem 24 - Calculus Early Transcendentals

Question

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 \(\rho(r, \theta, z)=5 e^{-r}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the limits of integration for r, θ, 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 \(2\)

Step 2: 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: $$M = \int_{0}^{2} \int_{0}^{2\pi} \int_{0}^{3} 5e^{-r} r \, dr \, d\theta \, dz$$

Step 3: Perform the r-integration

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

Step 4: Simplify the integrand

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

Step 5: Perform the θ-integration

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

Step 6: Perform the z-integration

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

Step 7: 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 \(20\pi-80\pi e^{-3}\).