Table of Contents

Calculating Volumes with Cylindrical Shells Examples 1
We will now look at some more examples of calculating volumes via the cylindrical shell method. We will use the formula $\int_{a}^{b} 2\pi x f(x) \: dx$ extensively.
Example 1
Calculate the volume of the solid by rotating the region between the functions $f(x) = x$ and $g(x) = x^2$ about the $y$axis.
The graph below depicts the region we're rotating about the yaxis:
Notice that we are essentially taking the volume obtained when rotating f(x) around the yaxis and subtracting the volume obtained when rotating g(x) around the yaxis.
First, let's find the points of intersection which are rather easy as f(x) = g(x) when x = 0 or x = 1. So the upper and lower bounds of our integral will be 0 and 1. Now applying our cylindrical shells formula we obtain:
(1)Example 2
Calculate the volume of the solid by rotating the region of area trapped between the function $f(x) = x  x^2$ and $g(x) = 0$ around the line $x = 2$.
This time we are rotating the region bounded by $f$ and the $x$axis along the line $x = 2$ as illustrated in this diagram:
This time the radius of our shells are $2  x$ as show by the graph above. Additionally, the function $f$ has $x$intercepts at $x = 0$ and $x = 1$, so the area trapped by the curve $f$ and the $x$axis is bounded on the interval $[0, 1]$. Hence applying the formula we get:
(2)Example 3
Calculate the volume obtained by rotating the region bounded by $xy = 1$, $x = 0$, $y = 1$, and $y = 3$ around the $x$axis using cylindrical shells.
We first note that we will be rotating the following region about the xaxis:
We note that this area is defined by taking the function $x = 1/y = y^{1}$ and integrating it on the interval 1 ≤ y ≤ 3. The radius of our shells will be [[$ y
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