. Let S be the solid obtained from rotating the region bounded by y = 2x and y = 2x 2 about y-axis. Find the volume of the solid S

Answer 1

Looking at the cross sections, we can see a "stack of washers" where the area of each washer is π(r_o² - r_i²) and the thickness is dy.

r_o is the x value of the y = 2x² function:

y/2 = x²

√y/2 = x = r_o

r_i is the x value of the y = 2x function:

y = 2x

y/2 = x = r_i

Area of washer: π[(√y/2)² - (y/2)²] = π(y/2 - y²/4) = π(2y/4 - y²/4) = π/4(2y - y²)

Volume of S: ₀∫²π/4(2y - y²)dy

V = π/4 ₀∫²(2y - y²)dy

V = π/4(y² - y³/3) evaluated between 0 and 2

V = π/4(2² - 2³/3)

V = π/4(4 - 8/3)

V = π/4(12/3 - 8/3)

V = π/4(4/3)

V = π/3 cubic units