Volume of solid of revolution: Disk Method: parabola about x axis:
Find the volume V generated by rotating the region in the first quadrant bounded by and about the x axis.
In other words, the parabola shown in the first figure is rotated about the x axis to generate the second figure.
We can use the narrow rectangle shown in the third figure to generate the thin disk shown in the fourth figure.
The volume, V is the sum of the volumes, dV each, of the disks.
The height, or y coordinate, of the top of the rectangle, is also the radius of the disk. (Even though the heights of the left and right sides of the rectangle are different in the diagram, the difference is negligible when we shrink the width, dx, for integrating.)
This radius, R, is .
The volume, dV, of the disk is
We now use integration to add up the volumes of the disks to get the total volume, V, of the figure of revolution:
We can multiply out the quadratic to get a sum of powers of x and then apply the Power Rule for integration: