On this page, you will find out how to find the volume of a solid with either a circular or semicircular base, and square cross sections.
Pro-tip: watch the YouTube video to really understand what’s going on.
Problem: Find the volume of a solid with a circular base, radius 4, and a cross section that is square.
If you slice perpendicular to base, you have a cross-section. These cross-sections are specified by the problem as squares.
So for a circular base with a radius 4, you have a cross-section or slice that is square as aforementioned. So to find the volume of the solid, you have to find the volume of the slice or cross-section.
Next you have to use the equation of a circle to find the height or length of one side of the square. This works because we are using a circular base.
We know that r is 4, solve for y:
y = (16-x2)1/2
So 2 of these give you the whole length of one side of the square: 2(16-x2)1/2
To find the area of the square cross-section/slice, you multiply 1 side by 1 side. All sides of the square are equidistant. So you end up with 4(16-x2)
All you have to do is add together all of the square cross section regions to find the volume of the solid. For this you use integration
We already know the interval is [-4, 4], because we are working with a circle:
@ interval [-4, 4], ∫ 4(16-x2)1/2dx
Note that dx is the thickness of the square cross section or slice.
Now you integrate to find the volume of the solid.
If you need the volume of a semicircular base, divide the volume of the circular base by 2 to acquire the semicircular solid volume.