Question

use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.

$x = 5y^{2}$, $y \geq 0$, $x = 5$; about $y = 2$

Explanation:

Step1: Find Intersection Point

To find the limits of integration, we find where \(x = 5y^{2}\) and \(x = 5\) intersect for \(y\geq0\). Set \(5y^{2}=5\), then \(y^{2} = 1\), so \(y = 1\) (since \(y\geq0\)). So \(y\) ranges from \(0\) to \(1\).

Step2: Recall Cylindrical Shells Formula (Horizontal Axis)

The formula for the volume using cylindrical shells when rotating about a horizontal axis \(y = k\) is \(V=\int_{a}^{b}2\pi\times(\text{radius})\times(\text{height})\,dy\). Here, the radius is the distance from \(y\) to the axis of rotation \(y = 2\), so \(\text{radius}=2 - y\). The height of each shell is the horizontal distance between the curves, which is \(5 - 5y^{2}\) (since \(x = 5\) is to the right of \(x = 5y^{2}\) for \(y\in[0,1]\)).

Step3: Set Up the Integral

Substitute the radius and height into the volume formula:

$$ V=\int_{0}^{1}2\pi(2 - y)(5 - 5y^{2})\,dy $$

First, expand the integrand:

$$
LATEXBLOCK0
$$

So the integral becomes:

$$ V = 2\pi\int_{0}^{1}(10-10y^{2}-5y + 5y^{3})\,dy $$

Step4: Integrate Term by Term

Integrate each term:

$$
LATEXBLOCK1
$$

Step5: Combine the Terms

Find a common denominator (12) to combine the fractions:

$$
LATEXBLOCK2
$$

Step6: Multiply by \(2\pi\)

Now multiply by \(2\pi\):

$$ V = 2\pi\times\frac{65}{12}=\frac{65\pi}{6} $$

Answer:

\(\boxed{\dfrac{65\pi}{6}}\)