Exercises

Please complete each exercise or answer the question before reviewing the posted solution comments.

1. You are working with a tank that has a radius that varies by height according to the following relationship (all measurements in feet): $$r(h)=h^2+\frac{1}{3}{h}^{\sqrt{3}}$$

   We can find the volume of a container with varying radius by using the following expression: $$V={\displaystyle \underset{0}{\overset{h}{\int }}\pi r}{(h)}^{2}dh$$

   Therefore, the integral we need to evaluate to find the volume of a tank that is H feet high and a radius that is a function of height according to r(h) above, is: $$V={{\displaystyle \underset{0}{\overset{H}{\int }}\pi \left(h^2+\frac{1}{3}{h}^{\sqrt{3}}\right)}}^{2}dh$$

   Use Matlab to determine the volume (evaluate the integral) for such a tank that is 5 feet high: $$V={{\displaystyle \underset{0}{\overset{5}{\int }}\pi \left(h^2+\frac{1}{3}{h}^{\sqrt{3}}\right)}}^{2}dh$$

   First, define a function that returns the cross-sectional area of the tank at a specified height.

             function xarea = tank_cross_section(h)
             % Returns the cross-sectional area of a tank
             xarea = pi*(h.^2+(1/3)*h.^sqrt(3)).^2;
           

   Notice the periods, this is so that the function still will work when a matrix is passed to it. This allows the quad function to have the function evaluate more than one value at a time. This ability is required for quad to correctly determine the result.

   Compute the value by typing:

             >> quad('tank_cross_section',0,5)
           

   Matlab will return with ans = 2.9652e+003 In other words, our tank volume is 2965.2 cubic feet. Remember, if you plan to use this value again, name it (such as TVol = quad('tank',0,5). That way, TVol = 2965.2)

   Note: Although it is good programming practice to define the function in a separate M-file, the quad function can be executed with the single line:

             >> TVol = quad('pi*(h.^2+(1/3)*h.^sqrt(3)).^2',0,5)