To Explain: to calculate the approximate volume of solid (S) by a Riemann sum.
To Write: An expression for the exact volume of solid.
Find the approximate volume of the ith strip (V):
Substitute for width of the strip and for the area of the strip.
The exact volume of the solid is the summation of the volume of all slabs.
Consider the number of “slabs”.
Find the exact volume of the solid as follows:
Rearrange Equation (1)
Here, the lower limit is a, upper limit is b, total area of the solid is A, and total width is x.
Thus, the exact volume of solid (S) is .
To Find: The cross-sectional areas if S is a solid of revolution, (S)
Consider the revolution solid is a washer shape. The expression for the area for the washer is as follows:
Here, is outer radius of washer and is inner radius of washer.
Consider the revolution solid is a disk shape. The expression for the area for the disk is as follows:
Here, r is radius of the disk.
Therefore, the cross-sectional area for revolution solid is defined by washer and disk method.
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