Problem 52 - Essential Calculus

Question

Find the volume of the torus generated by revolving the region bounded by the graph of the circle about the \(y\) -axis. $$ (x-h)^{2}+y^{2}=r^{2}, h>r $$

Step-by-Step Solution

Answer

The volume of the torus generated by revolving the circle \((x - h)^2 + y^2 = r^2\) about the y-axis can be calculated by the provided steps and gives: \(Volume = 2 \pi \int_{-r}^{r} y \pi (h - \sqrt{r^2 - y^2})^2 dy\). The final solution will depend on the specific values of \(h\) and \(r\).

Step 1: Identify and interpret key values

The equation of the circle is \((x - h)^2 + y^2 = r^2\), where \(h > r\). This indicates that the circle's center is at point \((h,0)\) and it has a radius of \(r\). It is detached from the y-axis. The circle is revolved around the y-axis to generate a torus.

Step 2: Setup the formula for volume of revolution

The torus can be viewed as an amalgamation of cylindrical slices or washers, each with volume \(V = 2 \pi \cdot y \cdot A\), where \(A\) is the cross-sectional area of each washer and \(y\) is the distance from the origin (0,0) to the centroid of the disk. Since the torus has an inner radius of \((h - r)\) and an outer radius of \((h + r)\), the radius of each washer is \(r = x = h - \sqrt{r^2 - y^2}\). So, \( A = \pi r^2 = \pi (h - \sqrt{r^2 - y^2})^2\).

Step 3: Integrate to find the volume

To calculate the total volume of the torus, integrate over the height of the torus. This means to integrate \(V = 2 \pi \cdot y \cdot A\) from \(y = -r\) to \(y = r\). This gives: \(Volume = 2 \pi \int_{-r}^{r} y \pi (h - \sqrt{r^2 - y^2})^2 dy\). Solving the integral will provide the volume of the torus.