Volume and the Trapezoidal Rule

The following is an exercise from Calculus , Hughes-Hallett et al.

Exercise 17, page 435. The hull of a certain boat has widths given by the following table. Reading across a row of the table gives widths at points 0,10,...,60 feet from the bow (front) to the stern (back) at a certain level below the waterline. Reading down a column of the table gives widths at levels 0,2,4,6,8,feet below waterline at a certain point from the bow to the stern. Use the trapezoidal rule to estimate the volume of the hull below waterline.

Boat Hull widths
Bow Stern
0 10 20 30 40 50 60
0 2 8 13 16 17 16 10
Depth 2 1 4 8 10 11 10 8
below 4 0 3 4 6 7 6 4
water 6 0 1 2 3 4 3 2
(ft.) 8 0 0 1 1 1 1 1

Can we use the table data to get an image of the boathull?

Using a powerful mathematical software package called MATLAB we can get the desired image assuming that the hull, at any given depth, is symmetric about its centerline. With this visual image we hope to formulate an approximation to the volume of the hull. Here's the sketch of the hull:

If you have MATLAB (or the student version) you can download the m-file in zipped form, boathull.m and the table data boathull.mat also in zipped form. Simply unzip these files and copy boathull.m and boathull.mat to the MATLAB directory where you store your work. Then type the command: boathull at the MATLAB prompt. Alternately, you can follow this link to an M-book that provides the coding for boathull.m. To view the file you need to have Microsoft Word (version 6.0 or higher) installed on your machine with your browser helper application for files of type *.doc set to Winword.exe.

Based on the table of widths we can also sketch the hull crossections at the various depths. For example the shape of the hull at four feet below the waterline is, based on the given data, something like:

We can also show the corresponding hull profiles as level curves at the depths corresponding to the table data (shown circled in the image)

How do we compute the volume of the hull ?

Now that we have some idea of the the hull's shape, how do we compute its volume? Let's begin by considering a typical crossection, say at depth = 2, of the hull. In the following picture we sketch half of the hull only. The region shaded green is a trapezoid whose area is easy to calculate.

The area of the green trapezoid is:

(average height) x (base) =((w4+w5)/2) 10.

Summing over all of the trapezoids gives the following expression for the Trapezoidal Rule :

half of crossectional area = 5(w1+2[w2+w3+w4+w5+w6]+w7)

But each height in the previous picture is one half the hull width and hence we find the following simple formula for the crossectional area at depth = 2

crossectional area = 5(w(2,1)+2[w(2,2)+w(2,3)+w(2,4)+w(2,5)+w(2,6)]+w(2,7)) = 475

where w(i,j) denotes the ith row and jth column entry of the original table of hull widths. We carry out the corresponding calculations for the remaining depths and find the following crossectional areas.

depth 0 2 4 6 8
area 760 475 280 140 45

Based on this table we can make a plot of crossectional area vs. depth below waterline. The shaded red region represents the hull volume between 2 and 4 feet below waterline.

Using the trapezoidal rule again, as indicated in the last picture, gives the computed volume: 2595 cubic feet.

If you want to see the solution presented from a different point of view, (that given in the Instructors Solutions Manual) click the button :