# Volume of a Solid ?

• Mar 23rd 2010, 11:41 AM
maxreality
Volume of a Solid ?
Find the volume of the solid generated by revolving around the x-axis
the region bounded by the x-axis, and one arch of the cycloid
x = t−sint, y = 1−cost.
where 0 ≤ t ≤ 2pi.
[ Hint : Use the disk method and dV = piy2dx = piy2(dx/dt) dt ]

I'm confused on this problem. I think it's because I don't actually know what they're asking... Is y the upper and x (t-sint) the width?
• Mar 23rd 2010, 01:53 PM
skeeter
Quote:

Originally Posted by maxreality
Find the volume of the solid generated by revolving around the x-axis
the region bounded by the x-axis, and one arch of the cycloid
x = t−sint, y = 1−cost.
where 0 ≤ t ≤ 2pi.
[ Hint : Use the disk method and dV = piy2dx = piy2(dx/dt) dt ]

I'm confused on this problem. I think it's because I don't actually know what they're asking... Is y the upper and x (t-sint) the width?

$\displaystyle dV = \pi y^2 \, dx$ is the volume of a representative disk in terms of y and x.

$\displaystyle dV = \pi y^2 \cdot \frac{dx}{dt} \, dt = \pi(1-\cos{t})^2 \cdot (1-\cos{t}) \, dt$ is the same thing in terms of t since y and x are given in terms of t

$\displaystyle V = \pi \int_0^{2\pi} (1-\cos{t})^3 \, dt$