Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the curves about the given axis. y = cos(πx/4), y = 0, 0 ≤ x ≤ 2; about the y-axis

Answer 1

Answer:

5.81

Step-by-step explanation:

The method of cylindrical shells uses this formula:

$\int\limits^a_b {2 \pi * shellRadius * shellHeight} \, dx$

It integrates the pieces of area in the region.

In this case:

$2 \pi * \int\limits^2_0 {x * cos(\frac{\pi}{4} * x)} \, dx$

Integrating by computer:

$2*\pi*(\frac{4}{\pi^2}*(\pi*x*sin(\frac{\pi}{4}*x) + 4*cos(\frac{\pi}{4}*x)))$

Evaluating between 0 and 2

$2\pi * (\frac{4}{\pi^2}*(\pi*2*sin(\frac{\pi}{4}*2) + 4*cos(\frac{\pi}{4}*2)) - \frac{4}{\pi^2}*(\pi*0*sin(\frac{\pi}{4}*0) + 4*cos(\frac{\pi}{4}*0))) = 5.81$