How to find the volume of a solid with triangle base & (semi-) circular cross sections

On this page, I will collect my notes and analysis that will help me find the volume of a solid with a triangle base and semi-circular cross sections.

Example problem

Use the general slicing method to find the volume of the solid whose base is the triangle with vertices(0,0)​, (6,0)​, and (0,6) and whose cross sections perpendicular to the base and parallel to the​ y-axis are semicircles. What is the volume?

The volume of a solid that extends from x = a to x = b and has a known integrable cross-sectional area A(x) perpendicular to the x-axis is given by the formula for the general slicing method:

The graph below shows the base of the solid, a triangle with vertices (0,0) (6,0) (0,6).

The diagram below shows a generic cross section of the solid, a semicircle that is perpendicular tot he base and parallel to the y-axis.

The red line segment shown in the graph below represents the diameter of a typical semicircle. In this case, the diameter varies with x along the hypotenuse of the triangle.

What is the diameter as a function of x? Answer: 6-x

Since the diameter is 6-x, the radius of the semicircle is then (1/2)*(6-x). Remember that the area of a semicircle is half the area of a circle: (1/2)*πr2, where r is the radius.

*If you are looking for the area of a full circle, then use that formula.

Now we need to find the cross-sectional area in terms of x.

A(x) = (1/2)*πr2   << Use the formula for the area of a semicircle

= (π/2)((1/2)*(6-x))2   << Substitute r = (1/2)*(6-x)

= (π/8)(6-x)2   << Simplify the coefficient

Next, you need to determine the limit of integration.

What is the lower limit, a? a = 0.

What is the upper limit, b? b = 6.

Now you can integrate A(x) from 0 to 6 to find the volume.

@ interval [0,6]; Volume = (π/8)(6-x)2 dx =

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