Question

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis. x = 3 + y2, x = 0, y = 2, y = 3 V = ___________

Answer 1

Answer:

Step-by-step explanation:

Let's give this a go here.  The volume formula for the shell method while rotating about a horizontal line is

$V=2\pi \int\limits^a_b {p(y)h(y)} \, dy$

where p(y) is the distance from the axis of rotation (the x-axis) to the center of the solid.  This is a positive distance and it is just y.

h(y) is the horizontal height of the function.  Our function starts at x = 0 and ends at the function itself, so h(y) = 3 + y^2.

In the shell method when rotating about a horizontal line, we need to use x = y equations, and y-intervals.  Setting up our integral then:

$V=2\pi \int\limits^3_2 {y(3+y^2)} \, dy$

We can simplify this a bit by distributing the y into the parenthesis:

$V=2\pi \int\limits^3_2 {3y+y^3} \, dy$

Integrating gives us

$V=2\pi|\frac{3y^2}{2}+\frac{y^4}{4}|$ from 2 to 3

Using the First Fundamental Theorem of Calculus:

$V=2\pi[\frac{135}{4}-\frac{40}{4}]$ which simplifies down to

$V=\frac{95\pi}{2}$