As the formula for finding the volume of the solid generated by revolving an area about the axis is the antiderivative of π*r^2.
Therefore, in the context of this problem, the expression for the volume would be:
However, as one of the integral limit is infinity, this is an example of an improper integral. Therefore, we should rewrite the integral as a limit, like this:
Limit as t approaches infinity,
Now, we can evaluate the integral using integration by parts. Before doing that, let's rewrite the integral into a new form like this:
Now, we can use the Tabular method to evaluate the integral by setting x^2 as u and e^-2xdx as dv.
u dv
x^2 e^-2x
2x -1/2 e^-2x
2 1/4 e^-2x
0 -1/8 e^-2x
Then, the expression of the volume would equal to:
(limit as t approaches infinity), V = π [ -1/2 * x^2 * e^-2x - 1/2 * x * e^-2x - 1/4 e^-2x] from 0 to t.
With this is mind, the volume should equal:
V = π [0 - 0 - 0 - 0 - 0 - (-1/4)] = π/4 .
Therefore, the final answer would be that the volume converges π/4 .
Hi,
Thank you for your question.
As the formula for finding the volume of the solid generated by revolving an area about the axis is the antiderivative of π*r^2.
Therefore, in the context of this problem, the expression for the volume would be:
However, as one of the integral limit is infinity, this is an example of an improper integral. Therefore, we should rewrite the integral as a limit, like this:
Limit as t approaches infinity,
Now, we can evaluate the integral using integration by parts. Before doing that, let's rewrite the integral into a new form like this:
Now, we can use the Tabular method to evaluate the integral by setting x^2 as u and e^-2xdx as dv.
u dv
x^2 e^-2x
2x -1/2 e^-2x
2 1/4 e^-2x
0 -1/8 e^-2x
Then, the expression of the volume would equal to:
(limit as t approaches infinity), V = π [ -1/2 * x^2 * e^-2x - 1/2 * x * e^-2x - 1/4 e^-2x] from 0 to t.
With this is mind, the volume should equal:
V = π [0 - 0 - 0 - 0 - 0 - (-1/4)] = π/4 .
Therefore, the final answer would be that the volume converges π/4 .
Does this answer your question?