Using the general slicing method to find the volume of a semi-circle whose cross sections are squares.
In finding the volume of a solid, described below, I was close in finding the equation, but neglected a coefficient. Please see the question below.
Use the general slicing method to find the volume of the following solid.
The solid with a semicircular base of radius 8 whose cross sections, perpendicular to the base and parallel to the diameter, are squares.
Place the semi-circle on the xy-plane so that its diameter on the x-axis and it is centered on the y-axis. Set up the intregral that gives the volume of the solid.
The resulting integral is:
$$ \int_{0}^{8}4(64-y^2)dy $$
The integral I came up with is: $$ \int_{0}^{8}(64-y^2)dy $$
Where is the 4 coming from? Why do I need to multiple the equation of the circle by 4?
Thanks for any help.
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$\begingroup$Since this has gotten bumped, here's a diagram for posterity, with a thousand words omitted.
While integrating from $ y=0 $ to $ y = 8 $ you should be considering double the y coordinate by symmetry of y-axis.
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