Calculate the volume of a variable trapezoid rotating around an axis

Question

I would like to calculate the volume of the section shown in blue in the 2 drawings below, which section results from the collision of two liquid jets. The base for the rotation is an ellipse as shown in the second drawing. To calculate the volume I believe I can rotate a variable area trapezoid (from point "S" to the ellipse edge as shown in the first figure) around a y-axis that runs through the point "S". "S" is not the center of the ellipse. "S" is the stagnation point caused by the collision of the two liquid jets and it is integral to the entire model. A different point cannot be selected and all equations presented below result from this stagnation point location. It also turns out that "S" is just beyond the left focus of the ellipse, but I don't think its position is that important, since the lengths ("p") from "S" are known. The third figure shows the particulars of the trapezoid that is to spin around the axis through "S" but along the base of the ellipse. So the formula for the area of the trapezoid is,

$$A = \frac{h_i+h_s}{2}p$$

where h is the length of the trapezoid base on each end and p is the "height" of the trapezoid. In reality "h" is a height, and "p" is more like a radius from the point "S", but for the trapezoid area I am using the definitions typically used. "p" varies in length with azimuthal angle as measured from the point "S".

For this particular situation, the following is the general relationship between h and p,

$$h = Ap$$

where "A" is a constant. Now as the rotation around "S" occurs it is important to understand that p varies in length which of course causes h to vary as well.

During the rotation $h_s$ is a constant value, so let's set it equal to the constant "B".

So,

$$Area = \frac{(Ap+B)}{2}p$$

So, since the area of the trapezoid is defined, I thought I could just use the following integral

$$V = \int_{0}^{2\pi}\frac{(Ap+B)}{2}pd\phi = \int_{0}^{2\pi}\frac{(Ap^2+Bp)}{2}d\phi $$

but I realize now, that since "p" has units of length, this calculation only gives me $length^2$ instead of $length^3$, which is volume. The constant "A" has no units and the constant "B" as defined above has units of length. I'm guessing I need to multiply by "p" but I'm not really understanding why. The reason I don't understand is because "p" in the area of the trapezoid is the complete length from S to the ellipse edge; hence I thought there was no need to integrate over the length p. I can see that simply using $2\pi$ doesn't give me a length so that is probably the issue. I guess I'm not clear what the variable is that is being integrated because the values of "p" are already the entire length from the point "S" to the ellipse edge.

As I think more about it, maybe the equation that I have only gives me the surface area of the shell that is produced by spinning around the edge. Maybe I need to have a double integral which includes dp in order to get the interior volume. Please confirm if my thinking is correct.

So, maybe $$V = \int_{0}^{2\pi}\int_{0}^{p}\frac{(Ap+B)}{2}pdpd\phi$$

Edit: But now again I am confused because the trapezoid area is for the entire trapezoid from point S to the edge of the ellipse.

Edit: I need to solve this using an integral method instead of something like Pappus's 2nd theorem.

Edit: "p" is a function of the angle, $\phi$, which is measured from the point "S". Also, I don't necessarily need the integral(s) to be solved. I need help setting them up and simplifying them as much as possible.

Answer 1

Since no responses have been posted on this question, here is what I currently think the solution is. I would very much appreciate comments on this solution. Please comment if you think this is the right way to handle the problem or if it is the wrong way to handle the problem.

First, I want to clear up something that might be confusing. The trapezoid shown as Fig. 3 in the original post revolves around a vertical line through "S". The reason that the path turns out to be elliptical and not circular is because "p" is a function of the azimuthal angle. It is largest at the azimuthal angle of $0$ degrees and smallest at the azimuthal angle of 180 degrees as shown in the first figure attached to this answer. This is what causes the path to look elliptical, not because an elliptical path has been forced. Because the height ("h") is directly proportional to "p" as given by

$$h=Ap$$

as "p" gets smaller, "h" does as well. This causes the trapezoid area to significantly change as it is rotated around the vertical line through "S".

My thought process behind this tentative answer is that the differential volume is equal to the area multiplied by the differential thickness. As shown in Fig. 2 below (in this answer), the differential distance that the trapezoid moves I believe is p$d\phi$. The trapezoid area is given by Eq. 3 in the question above,

$$Area = \frac{(Ap^2+Bp)}{2}$$

Thus the formula for the volume is,

$$V = \int_{0}^{2\pi}\frac{(Ap^2+Bp)}{2}pd\phi$$

noting that "p" is a function of $\phi$. Actually calculating the volume requires numerical methods as the relationship between "p" and $\phi$ is quite complex. But for the moment having this equation meets my needs.

Any comments on this method for this problem would be greatly appreciated.