A prolate spheroid (two out of three axes are equal, and less than the third) with an equaitorial radius (the length of the two equal axes) of 1 and a major axes(the horizontal length) of 12, has a segment with perpendicular planes cut out of it, such as is pictured above.

If the first cut is 5 units to the left of the origin and the second cut is 4 units to the left of the origin, what is the volume of the segment?

A sphere with a radius of 3 has a segment with perpendicular planes cut out of it, such as is pictured above.

If the first cut is 2 units to the left of the origin and the second cut is 1 unit to the right of the origin, what is the volume of the segment?

Finding area under a curve is extremely similar to finding the volume. The volume of a cylinder is . When finding the volume of a function rotated around the x-axis, we will look at summing infinitesimal cylinders (disks). The height of each of these cylinders is , the radius of the cylinder is the function since given any x value, f(x) is the distance from the x-axis to the curve. Thus if we want to find the volume of a function f(x) between \[a,b\] that is rotated about the x-axis we simply use the equation .

Find the volume of the solid obtained from rotating the function about the x-axis and bounded by the y-axis and .

A sphere with a radius of 7 has a segment with perpendicular planes cut out of it, such as is pictured above.

If the first cut is 6 units to the left of the origin and the second cut is 5 units to the left of the origin, what is the volume of the segment?

Find the volume of the function

revolved around the -axis on the interval .

A sphere with a radius of 10 has a segment with perpendicular planes cut out of it, such as is pictured above.

If the first cut is 4 units to the left of the origin and the second cut is 3 units to the right of the origin, what is the volume of the segment?

What is the volume inside the bowl , ? Hint: This is a solid of revolution about the z-axis with a radius of .

Suppose I want to construct a cylindrical container. It costs 5 dollars per square foot to construct the two circular ends and 2 dollars per square foot for the rounded side. If I have a budget of 100 dollars, what's the maximum volume possible for this container?

For any given geometric equation with one variable, volume over a given region is defined as the definite integral of the surface area over that specific region.

Given the equation for surface area of any sphere, , determine the volume of the piece of the sphere from to .

For this problem assume that volume is given by:

, where is the surface area function.

Given the formula for surface area of a structure is given by .

Determine the volume function of this structure.

For this problem assume that is not a function of . given below is a constant.