Answer to Question #176662 in Calculus for Joshua

Question

Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by y=7-x^2, x=-2, x=2 and the x-axis about the x-axis.

Answer 1

A graph of the function $y=7-x^2$ between $x=-2$ and $x=2$ has the form:

In case we rotate the plot around $x$-axis and then fix $x$, we will get a circle of the radius $R=7-x^2$. The area enclosed by it is $S=\pi R^2=\pi(7-x^2)^2$. In order to find the volume of the solid, we consider equal intervals $\Delta x_i=x_{i}-x_{i-1}, i=1,...,n$, where $-2=x_0<x_1<x_2<...<x_n=2$ and the following sum: $S_n=\sum_{i=0}^n\pi(7-x_i^2)^2\Delta x_i$. In case we take the limit $\Delta x_i\rightarrow0$ (it is equivalent to $n\rightarrow\infty$ ), we will receive the integral that provides the volume of the solid: $V=\pi\int_{-2}^2(7-x^2)^2dx=\pi(\int_{-2}^2(49-14x^2+x^4)dx)=\pi(49x-\frac{14}{3}x^3+\frac{1}{5}x^5)|_{-2}^2=\pi(196-\frac{14\cdot16}{3}+\frac{64}{5})=\frac{2012}{15}\pi$

Answer: The volume of the solid is $\frac{2012}{15}\pi$