If the region enclosed by the y-axis, the line y=2, and the curve y=3√x (cubed rt of x), is revolved about the y-axis, the volume of the solid generated is?

Question

If the region enclosed by the y-axis, the line y=2, and the curve y=3√x (cubed rt of x), is revolved about the y-axis, the volume of the solid generated is?
Can you solve without a calculator please and show work

Answer 1

The function should be called CUBE ROOT of x, not cube root.

 

Make a drawing of the function in the first quadrant; the line y=2 intersects the curve at (8,2).

When this is revolved about the y-axis, cross sections are circles of radius x.

x = y³

 

The volume of a very thin circular cross section is pi(x²)dy and the limit of the Riemann sum for volume is represented by V =pi times def integral, y ranging from 0 to 2, integrand y^6 dy.