Calculus Volumes of Revolution...AP Calc Exam in 7 days..need explanation?

Question

Find the volume of the solid that results when the region bounded by y=x^3, x=2, and the x-axis is revolved around the line x=2.

Thanks for you help. (Do not use shell method)

Answer 1

Draw the region. You can see that y goes from 0 to 8,and that x=2 is a boundary of the region. For a given value of y the area extends from 2 back to y^(1/3). So the volume is
π∫(0 to 8) (y^(1/3) - 2)^2 dy
= π∫(0 to 8) (y^(2/3) - 4y^(1/3) + 4) dy
= π [y^(5/3) / (5/3) - 4y^(4/3) / (4/3) + 4y][0 to 8]
= π [(2^5 (3/5) - 4.2^4 (3/4) + 32) - 0]
= π (96/5 - 48 + 32)
= 16π/5.

Answer 2

Hi there! I'm also taking the AP Calc Exam this Wednesday. Finding the area between curves is a crucial part of the exam.

Here's an awesome site that explains it pretty well..

http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/twocurves.html