Answer:
V=864π/5
The 2nd shaded picture matches our volume. And the 1st 3D picture maps the volume once rotated.
Explanation:
The idea is that you're revolving the graph around an axis and getting a would-be solid objective. Think of it as finding the volume of a cylinder but instead of straight edges going up the height, the height follows a portion of a graph. We take the area of the circle with a variable radius and plug it into the integral:
∫πr²dy/dx
You need to know what axis you're revolving around, determine the radius of the shape, then your height which will determine variable you're integrating with respect to.
- Your axis of revolution is the y axis.
- The radius here is going to be the distance from the axis of revolution to some arbitrary point on the curve, which is simply x.
- Your height is going to be some small y distance, so dy is what we're integrating w/r to and we're told to go from 0 to 6.
So with that we plug in the r and simplify as needed before integrating to get the answer.
My work is in the attachment. Comment with any questions.