## Calculating Volumes of Revolution using the Disc and Washer Methods This guide explains how to calculate the volume of a solid formed by revolving a region around the x-axis or y-axis using the disc and washer methods. ### The Disc Method The disc method is used when the region being revolved is bounded by the curve and the axis of revolution. The disc method works by summing the volumes of infinitesimally thin cylindrical discs. Imagine slicing the solid into many thin cylinders. Each cylinder has a volume given by: $$V_{cylinder} = \pi r^2 h$$ where: * $r$ is the radius of the disc (the distance from the axis of revolution to the curve) * $h$ is the height of the disc (a small change in x or y, denoted as $dx$ or $dy$) To find the total volume, we integrate the volume of these discs over the relevant interval. #### Revolving around the x-axis: Remember that when revolving around the x-axis, your radius and limits of integration should be in terms of x. The volume $V$ of the solid formed by revolving a region bounded by $y = f(x)$, the x-axis, and the lines $x = a$ and $x = b$ around the x-axis is: $$V = \pi \int_a^b [f(x)]^2 dx$$ Let's find the volume of the solid formed by revolving $y = \sqrt{x}$ around the x-axis from $x = 0$ to $x = 4$. Here, $r(x) = \sqrt{x}$. Therefore: $$V = \pi \int_0^4 (\sqrt{x})^2 dx = \pi \int_0^4 x dx = \pi \left[ \frac{x^2}{2} \right]_0^4 = \pi \left( \frac{16}{2} - 0 \right) = 8\pi$$ #### Revolving around the y-axis: When revolving around the y-axis, your radius and limits of integration should be in terms of y. The volume $V$ of the solid formed by revolving a region bounded by $x = g(y)$, the y-axis, and the lines $y = c$ and $y = d$ around the y-axis is: $$V = \pi \int_c^d [g(y)]^2 dy$$ Let's find the volume of the solid formed by revolving $y = x^2$ around the y-axis, bounded by $x = 0$ and $y = 4$. First, solve for x in terms of y: $x = \sqrt{y}$. Then: $$V = \pi \int_0^4 (\sqrt{y})^2 dy = \pi \int_0^4 y dy = \pi \left[ \frac{y^2}{2} \right]_0^4 = \pi \left( \frac{16}{2} - 0 \right) = 8\pi$$ ### The Washer Method The washer method is used when the region being revolved is bounded by two curves and the axis of revolution. It's essentially the disc method with a hole in the middle. The washer method is a generalization of the disc method. When revolving a region between two curves, $f(x)$ and $g(x)$, around the x-axis, the volume of each washer is: $$V_{washer} = \pi ([f(x)]^2 - [g(x)]^2) dx$$ The total volume is found by integrating this expression. #### Revolving around the x-axis: The volume $V$ of the solid formed by revolving the region between $y = f(x)$ and $y = g(x)$ ($f(x) \ge g(x)$) around the x-axis from $x = a$ to $x = b$ is: $$V = \pi \int_a^b ([f(x)]^2 - [g(x)]^2) dx$$ #### Revolving around the y-axis: Similarly, for revolution around the y-axis, the volume is: $$V = \pi \int_c^d ([f(y)]^2 - [g(y)]^2) dy$$ A common mistake is forgetting to square the radius functions before integrating. Always remember that the area of a disc is $\pi r^2$. The choice between the disc and washer methods depends on the shape of the region being revolved and the axis of revolution. If the region is bounded by a single curve and the axis, use the disc method. If it's bounded by two curves and the axis, use the washer method. Consider the region bounded by $y = x$ and $y = x^2$ from $x = 0$ to $x = 1$. Revolving around the x-axis, the volume is: $$V = \pi \int_0^1 (x^2 - (x^2)^2) dx = \pi \int_0^1 (x^2 - x^4) dx = \pi \left[ \frac{x^3}{3} - \frac{x^5}{5} \right]_0^1 = \pi \left( \frac{1}{3} - \frac{1}{5} \right) = \frac{2\pi}{15}$$ ### Practice Problems 1. Find the volume of the solid generated by revolving $y = 1/x$ about the x-axis, bounded by $x = 1$ and $x = 3$. 2. Find the volume of the solid generated by revolving $y = x^{2/3}$ about the y-axis, bounded by $x = 0$ and $y = 1$. 3. Find the volume of the solid generated by revolving the region between $y = x$ and $y = x^3$ about the x-axis from $x = 0$ to $x = 1$. Remember to always sketch the region to visualize the solid and determine the appropriate radius function(s) and limits of integration.