A storage box has a base area of 314.5 square inches and a height of 8.5 inches. What is the volume of 5 of these boxes. Enter y
2 answers:
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Direct computation:
Parameterize the top part of the circle by
with , and the line segment by
with . Then
Using the fundamental theorem of calculus:
The integral can be written as
If there happens to be a scalar function such that
, then
is conservative and the integral is path-independent, so we only need to worry about the value of
at the path's endpoints.
This requires
So we have
which means is indeed conservative. By the fundamental theorem, we have