Solved examples: disk method volume of revolution y-axis

📚 Quick Study Guide

• 📐 The disk method is used to find the volume of a solid of revolution when rotating a function around an axis.
• 🔄 When rotating around the y-axis, we integrate with respect to $y$.
• ✍️ The formula for the disk method when rotating around the y-axis is: $V = \pi \int_{c}^{d} [x(y)]^2 dy$, where $x(y)$ is the function rewritten in terms of $y$, and $c$ and $d$ are the limits of integration along the y-axis.
• 💡 Remember to solve for $x$ in terms of $y$ before applying the formula.
• 🧮 Ensure the limits of integration ($c$ and $d$) are y-values.

Practice Quiz

1. What is the correct integral setup for finding the volume of the region bounded by $x = \sqrt{y}$, $y=0$, and $y=4$ rotated about the y-axis?
   1. $V = \pi \int_{0}^{4} y dy$
   2. $V = \pi \int_{0}^{2} x^4 dx$
   3. $V = \pi \int_{0}^{4} y^2 dy$
   4. $V = \pi \int_{0}^{2} x dx$
2. Find the volume of the solid generated when the region bounded by $x = y^2$, $x=0$, $y=1$, and $y=2$ is revolved about the y-axis.
   1. $\frac{31\pi}{5}$
   2. $\frac{\pi}{5}$
   3. $\frac{124\pi}{5}$
   4. $\frac{3\pi}{5}$
3. What adjustment needs to be made if rotating the region bounded by $x = f(y)$ and $x = g(y)$ where $f(y) > g(y)$ about the y-axis?
   1. The formula becomes $V = \pi \int [f(y) - g(y)]^2 dy$.
   2. The formula becomes $V = \pi \int [f(y)^2 - g(y)^2] dy$.
   3. The formula becomes $V = \pi \int [g(y) - f(y)]^2 dy$.
   4. The formula becomes $V = \pi \int [g(y)^2 - f(y)^2] dy$.
4. The region enclosed by $x = 0$, $y = 1$, $y = 4$, and $x = y^{3/2}$ is revolved about the y-axis. What is the volume of the solid generated?
   1. $\frac{255\pi}{2}$
   2. $\frac{51\pi}{5}$
   3. $\frac{255\pi}{5}$
   4. $\frac{51\pi}{2}$
5. Consider the region bounded by $x=\frac{2}{y}$, $y=2$, $y=6$ and $x=0$. Which integral represents the volume when rotated around the y-axis?
   1. $\pi \int_{2}^{6} (\frac{4}{y^2}) dy$
   2. $\pi \int_{2}^{6} (\frac{2}{y}) dy$
   3. $\pi \int_{0}^{2} (\frac{4}{y^2}) dy$
   4. $\pi \int_{0}^{2} (\frac{2}{y}) dy$
6. What is the volume generated by rotating the curve $x = \sqrt{2y - y^2}$ about the y-axis?
   1. $2\pi$
   2. $4\pi$
   3. $\pi$
   4. $8\pi$
7. If the region bounded by $x = e^y$, $y=0$, $y=1$ and $x=0$ is rotated around the y-axis, the integral setup is...?
   1. $\pi \int_{0}^{1} e^{2y} dy$
   2. $\pi \int_{0}^{1} e^{y} dy$
   3. $\pi \int_{0}^{1} e^{-y} dy$
   4. $\pi \int_{0}^{1} e^{-2y} dy$