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
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:
We can simplify this a bit by distributing the y into the parenthesis:
Integrating gives us
from 2 to 3
Using the First Fundamental Theorem of Calculus:
which simplifies down to