Finding the volume of a solid

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 1/25 x^2, x = 5, y = 0; about the y-axis

Answer 1

Use the shell method.

The volume of the cylindrical shell generated by a vertical cross section of the region at a distance of x from the y-axis is: 2πx(1/25)x²dx.

So, the volume of the solid is (2π/25)∫_{(0 to 5)}x³dx

= (2π/25)[(1/4)x⁴]_{(from 0 to 5)}

= (2π/25)(1/4)(625) =125π/2