Question

Find the volume V of the solid obtained by rotating the region bounded by the curves x = 2/3y, x = 0, y = 7 about the y-axis.

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curves x = 2/3y, x = 0, y = 7 about the y-axis, we use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves x = 2/3y, x = 0, y = 7 about the y-axis, we will use the method of cylindrical shells.

1. First, solve the equation x = 2/3y to find the limits of integration. When y = 7, x = 2/3 * 7 = 14/3.
2. Next, set up the integral to find the volume. The integral for the volume can be written as V = ∫(2πx)(dy) from y = 0 to y = 7.
3. Simplify the integral and integrate to find the volume. V = ∫(2π(2/3y))(dy) from y = 0 to y = 7. V = 2π∫(2/3y)(dy) from y = 0 to y = 7. V = 2π(1/3)y^2 from y = 0 to y = 7. V = 2π(1/3)(7)^2 - 2π(1/3)(0)^2. V = 2π(1/3)(49) - 0. V = 98π/3 cubic units.

Therefore, the volume of the solid is 98π/3 cubic units.