Calculating the Volume of a Bundt Pan
I decided to figure out the volume of my bundt pan. It has an inside diameter of 2", an outside diameter of 9", and a maximum depth of 3.5". The profile of the pan seemed to be parabolic, so I assumed it was a parabola.
Above is a section view of one side of the pan, with the pan being inverted. Using the dimensions of mine yields the points (1,0), (2.75, 3.5), and (4.5,0). The first and last points are where it crosses the x-axis, and the middle point is the maximum depth.
Since the curve is a parabola it will have the general form y= a*x^2 + b*x +c. To find the coefficients an augmented matrix of coefficients is set up. Using these points yields the following matrix:
Since the curve is a parabola it will have the general form y= a*x^2 + b*x +c. To find the coefficients an augmented matrix of coefficients is set up. Using these points yields the following matrix:
The values in the first row are from the first point, (1,0). The first term is 1^0, the second is 1^1, the third is 1^2, and the final term is the y value. The other two rows are found the same way, but uses the x and y values of the other two points.
Next the matrix is manipulated to its reduced echelon form using the Gauss-Jordan method. The result is the following matrix:
Next the matrix is manipulated to its reduced echelon form using the Gauss-Jordan method. The result is the following matrix:
The last column is the coefficients c, b, and a (from the top down). Therefore, the equation of the curve is:
Notice that this is the equation used to plot the graph above.
The next problem is to find the volume. To do this it must be imagined that the curve is rotated about the y-axis, and the volume will be the space between the rotated curve and the x-y plane. The shell method is the most convenient method to determine the volume. The integral for the shell method is:
The next problem is to find the volume. To do this it must be imagined that the curve is rotated about the y-axis, and the volume will be the space between the rotated curve and the x-y plane. The shell method is the most convenient method to determine the volume. The integral for the shell method is:
In this example h(x) = y, and p(x) is the distance from the y-axis to the differential element. Here p(x) = x. Substituting the values of h(x) and p(x) into the integral and multiplying it out gives:
Integrating gives the equation:
The values at the right are the boundaries. They are 1 and 4.5, even though the 1 looks like .1. Next the first theorem of calculus is used:
So the volume of the bundt pan is 141 cubic inches.