How do you determine the volume of a solid created by revolving a function around an axis?

Question

Cameron Alexander · Accepted Answer

#"Volume" = pi int_a^b f(x)^2 dx#

Given a function #f(x)# and an interval #[a, b]# we can think of the solid formed by revolving the graph of #f(x)# around the #x# axis as a horizontal stack of an infinite number of infinitesimally thin disks, each of radius #f(x)#.

The area of a circle is #pir^2#, so the area of the circle at a point #x# will be #pi f(x)^2#.

The volume of the solid is then the infinite sum of the infinitesimally thin disks over the interval #[a, b]#

So:

#"Volume" = int_a^b pif(x)^2 dx = pi int_a^b f(x)^2 dx#

Paisley Johnson · Answer

undefined To determine the volume of a solid created by revolving a function around an axis, you can use the method of cylindrical shells or the method of disks and washers, depending on the shape of the solid and the axis of rotation. Both methods involve integrating the cross-sectional area of the solid perpendicular to the axis of rotation along the interval of integration. This area is then multiplied by the thickness of the shell or disk and integrated over the appropriate interval to find the volume of the solid.