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 =
