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Use the general slicing method to find the volume of the following solid.
The solid with a semicircular base of radius 11 whose cross sections perpendicular to the base and parallel to the diameter are squares. Place the semicircle on the xy-plane so that its diameter is on the x-axis and it is centered on the y-axis. Set up the integral that gives the volume of the solid. Use increasing limits of integration.

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josealejandrocl

Answer:

The volume is 1789.33

Step-by-step explanation:

For a circle of radius 11, we have the following equation:

[tex]x^{2} +y^{2} =11^2=121[/tex]

Now, making it explicit for x:

[tex]x= \sqrt[]{121-y^2}[/tex]

Then, if we consider that for a height y, the length x is double, we have that the length of each cross section is given by:

[tex]s= 2\sqrt[]{81-y^2}[/tex]

With which, we can propose the following integral to obtain the volume that they are asking us:

[tex]\int\limits^{11}_0 {s^2} \, dy\\ \int\limits^{11}_0 {(2\sqrt{121-y^2} )^2} \, dy\\ \int\limits^{11}_0 {4*({121-y^2)}} \, dy\\4(121y-\frac{y^3}{3})[/tex] (evaluated between 0 and nine )

Finally, calculating, we have that the volume is V=1789.33

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