Question

Use the disk method to find the volume of the solid generated when the region bounded by y equals 11 sine x and y equals 0​, for 0 less than or equals x less than or equals pi​, is revolved about the​ x-axis. (recall that sine squared x equals one half left parenthesis 1 minus cosine 2 x right parenthesis ​.)

Answer 1

Answer:

  (121/2)π² ≈ 597.111 cubic units

Step-by-step explanation:

The volume is the integral of a differential of volume over a suitable domain. The problem statement tells us that the differential of volume should be a disk, so we have ...

  dV = A·dx = πy²·dx = π·(11sin(x))²dx = 121π/2(1 -cos(2x))dx

Then the integral is ...

$\displaystyle V=\int_{0}^{\pi}\frac{121\pi}{2}(1-\cos{(2x)})\,dx=\frac{121\pi}{2}\left(\int_{0}^{\pi}1\,dx-\int_{0}^{\pi}\cos{(2x)}\,dx\right)\\\\=\frac{121\pi}{2}(\pi -0)=\frac{121\pi^2}{2}$

The volume of the solid is 60.5π² cubic units.