Question

Find the volume of the solid generated by rotating the region bounded by y = sin(x), y = 0, x = 0, and x = π/2, about the x-axis.

Answer 1

Explanation:

V = (2π - 2) units^3

So, the volume of the solid is (2π - 2) units^3.

Answer 2

Explanation:

To find the volume of this solid, we will use the method of cylindrical shells. We will integrate the cross-sectional area of a cylinder, given by πr^2, over the height of the solid. The height of the solid is given by the difference between the upper and lower bounds of the region, which are y = sin(x) and y = 0, respectively. The radius of each cylindrical shell is given by the height y = sin(x).

So, the volume of the solid is given by the integral:

V = ∫[0, π/2] π * [sin(x)]^2 dx

Using the antiderivative of π * sin^2(x), which is -π * cos(x), we can find the definite integral:

V = -π * [cos(x)] from 0 to π/2

Evaluating the definite integral and simplifying, we get:

V = (2π - 2) units^3

So, the volume of the solid is (2π - 2) units^3.