How do you find the volume of the solid generated by revolving the graph of a function #f(x)# around a point on the x-axis?

Question

Scarlett Allison · Accepted Answer

#V = pi int (f(x))^2 d x#, between the given limits for x, around the point

#V = pi int (f(x))^2 d x#, between the given limits for x, around the

point.

This formula is based on

# triangle x to 0# of #sum F(x) triangle x=int F(x) d x,# over the

given range for x.

Here, an elementary area , in the form of a rectangle of length

f(x) and width #triangle x#, is revolved about its base on the x-axis,

to generate an elementary solid of revolution that is in the form of a

circular disc of radius f(x) and thickness #triangle x#. This elementary

volume for summation is

#triangle V=pi (f(x))^2 triangle x#.

Then, it is summation of the infinite series for V, in the limit.

#V = lim triangle V to 0# of #sum triangle V#

#= lim triangle x to 0# of #sum pi (f(x))^2 triangle x#

#=pi int (f(x))^2 d x#, over the given range for x.

Scarlett Allison · Answer

undefined To find the volume of the solid generated by revolving the graph of a function $ f(x) $ around a point on the x-axis, you can use the disk method or the washer method.

Disk Method:
1. Determine the axis of rotation and the limits of integration.
2. Slice the solid into thin disks perpendicular to the axis of rotation.
3. Express the volume of each disk as $ V = \pi (f(x))^2 \Delta x $.
4. Integrate the expression $ \pi (f(x))^2 $ with respect to $ x $ over the given bounds of integration.

Washer Method:
1. Determine the axis of rotation and the limits of integration.
2. Slice the solid into thin washers perpendicular to the axis of rotation.
3. Express the volume of each washer as $ V = \pi ((\text{outer radius})^2 - (\text{inner radius})^2) \Delta x $.
4. Integrate the expression $ \pi ((\text{outer radius})^2 - (\text{inner radius})^2) $ with respect to $ x $ over the given bounds of integration.

After setting up the integral using either the disk method or the washer method, evaluate the integral to find the volume of the solid.