Finding the General Equation for the Volume of a Bundt Pan
The cross section of a bundt pan is roughly a parabola. To determine the volume we must first find the equation that defines the parabola, and then find the volume by using the y-axis as the axis of revolution for the solid.
The first thing to do is to establish the points needed. Here is an approximate sectional view of the pan upside-down:
Finding the Coefficients
The top of the pan corresponds with the x-axis, and the center of the pan is on the y-axis. The point where the small radius intersects the top is at the point (A,0), where the large radius intersects the top is at (D,0), and the maximum depth is at (B,C).
The equation of a parabola is
, so we have to use a matrix to determine the coefficients.
, so we have to use a matrix to determine the coefficients.
Now we start reducing the matrix [additions (multi-line operations) are highlighted in yellow, and multiplications (single line operations)in blue]:
From this matrix we get the coefficients, and this is equation for the profile:
Finding the Volume
Instead of trying to use the above coefficients, we'll use the general equation
for the volume. The section is rotated about the y-axis, and the shell method is the easiest method to use. The volume is found using the integral:
where R is the large radius and r is the small radius. h(x) is the height of the diferential element, so h(x) = y. p(x) is the radius of gyration (the distance from the y-axis to the diferential element), so p(x) = x. Substituting these values into the integral gives:
Substituting the coefficients into this equation gives:
Substituting the values A, B, C, D, R, and r (as defined previously) into the above equation will give the volume of any bundt pan.