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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 y-axis:
Notice that we are essentially taking the volume obtained when rotating f(x) around the y-axis and subtracting the volume obtained when rotating g(x) around the y-axis.
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 x-axis:
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
(3)