volume in cube

Question

 Use your calculator to find the approximate volume in cubic units of the solid created when the region under the curve y = sin(x) on the interval [0, π] is rotated around the x-axis. (4 points)

4.93

1.57

6.28

2.00

Answer 1

Let's slice this using disks.

 

Disks have radius r = sin x and thickness dx. Their volume is dV = π sin²x dx

 

Thus the solid has volume:

 

∫dV = ∫₀^π π sin²x dx = ∫₀^π [π(1 - cos 2x)/2] dx = [πx/2 - π(sin 2x)/4]₀^π = π²/2 ≈ 4.93