Question

Sketch the region bounded by the curves y = 2x√x,y = 3, and y = 2/3 then find the volume of the solid generated by revolving the region about the x-axis.

a) 16π
b) 12π
c) 8π
d) 4π

Answer 1

To find the volume of the solid generated by revolving the region bounded by the curves about the x-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the curves about the x-axis, we can use the method of cylindrical shells. First, let's find the points of intersection between the curves. Set the equations equal to each other:
2x√x = 3
Solving this equation, we get x = 27/8. Now, we can set up the integral to calculate the volume of the solid:

V = ∫27/832πx(3-(2/3))dx

Simplifying, we have:
V = 2π∫27/83 x(5/3) dx

Integrating, we get:
V = 2π(5/3) ∫27/83 x dx

By evaluating this integral, we find the volume to be 12π.