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.
$$
S : \frac{x^2}{4} + \frac{y^2}{9} = 1
$$
Because the cross sections are isosceles right triangles with hypotenuse in the base, the length of the hypotenuse is $ 6\sqrt{1-\frac{x^2}{4}}$, which means that the area of the cross section is $9(1-\frac{x^2}{4})$
Best Answer
Since this has gotten bumped, here's a diagram for posterity, with a thousand words omitted.