Find the volume V of the solid obtained by rotating the region enclosed by the graphs .

Question

Find the volume V of the solid obtained by rotating the region enclosed by the graphs

x = sqrt y and x = 0 about the y-axis between y = 1 and y = 8.

Answer 1

the shape will be a tall bowl figure with a flat bottom.

This is the 2D shape in question being rotated

the discs will be horizontal so the integration variable will be y

INT(Pi*r^2)dy - y=1..8

INT(Pi*(sqrt(y)^2)dy

Pi*INT(y)

Pi*y^2/2 y=1..8

Pi*32-Pi/2 = 63Pi/2