Concept explainers
Describe the method of slicing for finding volumes, and use that method to derive an
The method of slicing for finding volumes, and use that method to derive an integral formula for finding volumes by the method of disks.
Answer to Problem 1RE
The formula for volume by slicing method is
The formula for volume by the method of disks is
Explanation of Solution
Slicing method:
In this method, the solid is divided into thin slabs. Then, the volume of each slab is approximated. The approximations are added to form a Riemann sum and the limit of the Riemann sum is taken to produce an integral for the volume.
Let
Let the cross-section area
Now, divide the solid into
Consider the width of
Where,
The approximations are added to form a Riemann sum
Take the limit of the Riemann sum to get the volume by slicing method as,
Therefore, the formula for volume by slicing method is
Finding volumes by the method of disks:
The method of finding the volumes of the solid with the disk-shaped cross sections is called the method of disks.
Let
Now, find the solid of revolution that is generated by revolving
The cross section of the solid taken perpendicular to the
The area of the region is
Therefore, the formula for volume by the method of disks is
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