Let region R be bounded by f(x)=sin x, the x-axis and the line x = π/2.

Question

Find the volume of the solid formed when region R is rotated about the y-axis.

Answer 1

Using Shell method volume is V = 2π∫₀^π/2xsinxdx = 2π(- xcosx + sinx)₀^π/2 = 2π

Answer 2

πMaking a sketch would look like this:

where the "washer" in the lower drawing is a cross section of the solid. To calculate the volume, you would create an equation for the volume of that washer, where "dy" is the thickness of the washer and π(r_o² - r_i²) is the area of the flat surface of the "washer" then you would integrate, or add up, all the washers from 0 to 1.

V = ₀∫¹π(r_o² - r_i²) dy

But r_o is always π/2 and r_i is the x-value of the function. The x-value of the function can be found using the inverse sine:

y = sin(x)

sin^-1(y) = sin^-1(sin(x)

sin^-1(y) = x

So, substituting we get:

V = ₀∫¹π((π/2)² - (sin^-1(y))²) dy

This is something you can just plug in a calculator to find the results to.

V = 2π cubic units