Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves x=4y2−y3 and x=0 about the x-axis.

Answer 1

Answer = 321.7

Explanation:

radius = y

height = $4y^{2} - y^{3}$

area of cylinder = 2π*r*h

Integrate the area to calculate the volume:

$= \int\limits^0_4 {2\pi(4y^{3} -y^{4}) } \, dy$

$= 2\pi (\int\limits^0_4 {4y^{3}dy -\int\limits^0_4 y^{4} dy )} \,$

$= 2\pi (256-\frac{1024}{5} )$

$=\frac{512\pi }{5}$

$= 321.7$