Calculus Solution (part) to the tank problem

Consider a right cylinder tank (end view shown to the left).
Develop a function which describes the volume of the tank as a function of the fluid height (x).

We establish a coordinate axis system shown to the left with the cylinder end centered at (h,k) and having a radius of R. The fluid height is the variable x. Clearly, 1/2 of the tank end area (the solid area) is obtained by integrating the upper portion of the circle equation from 0 to 2R. Since the circle is symmetrical, just multiply this area by 2 to obtain the entire area at any x.

Equation of a circle:

For this problem:

Solving the circle equation for y we any obtain the functions which create the top and bottom semi-circles:

Using just the function representing the upper portion of the circle:

Just for fun, let's do a numerical solution first before attempting an exact analytical solution using calculus.

Numerically integrating f(x) over the domain:

using Mathcad's syntax:

A graph of f(x) and Area(x) is shown to the left where:

This graph makes sense since the end area of the right cylinder we know is just:

and

Area(x) increases slowly at the bottom and top of the tank as expected (WHY?).

Now let's do an analytical calculus solution. We know:

Integrating f(x) yields the area

The 1st indefinite integral

Where C is some constant of integration to be determined later.

The 2nd indefinite integral

We know from the CRC (copywrite 1971) page 415, Calculus Integral Equation # 242:

where:

Also from the CRC on page 414, equation 238 (when c < 0)

So putting it all together:

Which simplifies to:

Remembering for our problem:

Now when x = 0, the area = 0. So, we can solve for C

which really is exactly:

or

The area (just written as a function F(x)) is:

and graphed over the domain:

Let's include the area function that was previously derived numerically:

Since the analytically computed area function [F(x)] overlays the area numerically computed [Area(x)], the function F(x) is correct and equal to Area(x).

Volume for a right cylinder tank would be easy to compute since
Volume = (End Area)(Length).

The analytical end area function is therefore:

where C is :