Disk method y-axis interactive quiz and solutions

📚 Quick Study Guide

• 🔄 Disk Method Basics: When revolving a region around the y-axis, your integral will be with respect to $y$. This means you need to express your function as $x = f(y)$.
• 📐 Formula: The volume $V$ is given by the integral $V = \pi \int_{c}^{d} [f(y)]^2 dy$, where $c$ and $d$ are the limits of integration along the y-axis.
• 🧭 Limits of Integration: Determine the correct limits $c$ and $d$ by finding the y-values where the region starts and ends.
• 🧮 Function Squared: Make sure to square the function $f(y)$ before integrating.
• ✏️ Integration: Evaluate the definite integral to find the volume.

Practice Quiz

1. What is the correct formula for the disk method when revolving around the y-axis?
   1. $V = \pi \int_{a}^{b} [f(x)]^2 dx$
   2. $V = \pi \int_{c}^{d} [f(y)]^2 dy$
   3. $V = 2\pi \int_{a}^{b} x f(x) dx$
   4. $V = 2\pi \int_{c}^{d} y f(y) dy$
2. If the region is bounded by $x = y^2$, $y = 0$, and $y = 2$, what integral represents the volume when revolved around the y-axis?
   1. $\pi \int_{0}^{2} y^2 dy$
   2. $\pi \int_{0}^{2} y^4 dy$
   3. $\pi \int_{0}^{4} x^2 dx$
   4. $\pi \int_{0}^{4} \sqrt{x} dx$
3. The region bounded by $x = \sqrt{y}$, $y = 1$, and $y = 4$ is revolved around the y-axis. What is the volume?
   1. $\frac{15\pi}{2}$
   2. $\frac{3\pi}{2}$
   3. $\frac{15}{2}$
   4. $15\pi$
4. Given $x = 3y - y^2$ and the y-axis bounds $y=0$ and $y=3$, what is the volume when the region is rotated around the y-axis?
   1. $\frac{81\pi}{5}$
   2. $\frac{243\pi}{5}$
   3. $\frac{81}{5}$
   4. $81\pi$
5. Consider the region enclosed by $x = y^3$, $y = 1$, and $x = 0$. Find the volume of the solid generated when this region is revolved about the y-axis.
   1. $\frac{\pi}{7}$
   2. $\frac{2\pi}{7}$
   3. $\frac{\pi}{3}$
   4. $\frac{2\pi}{3}$
6. The area enclosed by $x = 4 - y^2$ and the y-axis is rotated about the y-axis. What is the volume of the resulting solid?
   1. $\frac{256\pi}{15}$
   2. $\frac{64\pi}{3}$
   3. $\frac{256}{15}$
   4. $\frac{64}{3}$
7. If the region is bounded by $x = e^y$, $y = 0$, $y = 1$, and $x = 0$ is revolved around the y-axis, what is the volume?
   1. $\frac{\pi}{2}(e^2 - 1)$
   2. $\pi(e - 1)$
   3. $\frac{1}{2}(e^2 - 1)$
   4. $\pi e^2$