Question

Find the volume V obtained by rotating the region bounded by the curves y = sin(x), y = 0, 0 ≤ x ≤ π/2 about the x-axis?

Answer 1

To find the volume obtained by rotating the region bounded by the curves y = sin(x), y = 0, 0 ≤ x ≤ π/2 about the x-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume obtained by rotating the region bounded by the curves y = sin(x), y = 0, 0 ≤ x ≤ π/2 about the x-axis, we need to use the method of cylindrical shells.

1. First, let's set up the integral for the volume:
2. V = ∫(0 to π/2) 2πx(f(x))dx, where f(x) is the top curve (y = sin(x)) and x is the variable of integration.
3. Next, we express sin(x) in terms of y:
4. x = arcsin(y)
5. Substitute this into the integral:
6. V = ∫(0 to 1) 2πarcsin(y)y dy
7. Integrate using substitution or integration by parts to obtain the volume