**Calculus Homework Help**

**Introduction**

Intent. The pedagogical goal of this project is for students to use numerical integration techniques to solve an engineering design question that cannot be tackled through any analytic means. It is meant to be an exercise that requires a bit of creativity, along with the ability to apply “integration” on a problem that you have not fully practiced in homework, asking you to read, understand, and design.

## Problem description: Designing a pirate cave door

You have been hired by Bill Johnson to design the door to an underwater pirates cave that is being constructed. Bill wants the door to be in the shape of a skull and has provided a hand drawn sketch to give the basic idea. (See Figure 1.) Bill has already decided that the door will be 8 cm thick and constructed from stainless steel. The additional specifications are related to the shape:

**Figure 1: Client provided sketch of the desired door, with reference line at top indicating the waterline.**

- The door will be in the shape of a skull. • The door must be between 3 m and 4 m in height. • The door must be at least 2 m wide.

The top of the door will be 2 m below the surface of the water. The door will be mounted on a pivoting point at the center of pressure, so that it can be easily opened even with sea pressure against the door.

In addition to providing a scaled drawing of the door shape, you are also required to determine:

- The total hydrostatic force on the door, so that the door mount can be properly designed.
- The location of the center of pressure, so that the pivot can be placed at the appropriate location.

You should present your findings in a short technical report.

#### Mathematical and Physics (hydrostatics) background information.

In Section 8.3 of the book, we worked on problems of hydrostatic force. So, computing that force is something that you (in principle) already know how to do. Recall that hydrostatic pressure at a distance x meters below the waterline is given by

P(x) = ρgx, (1)

where ρ is density, and g is acceleration due to gravity.

If we denote by w(x) the width of the plate at depth x, then we compute the total force on the plate by integration:

Figure 2: On an arbitrary plate positioned vertically, the width (w(x)) is a function of position.

FHS = ∫ b

a P(x)w(x)dx. (2)

The center of pressure is the point where the total sum of a pres- sure field acts on a body. If the pressure is constant across the body, then the center of pressure coincides with the centroid. However, in problems such as this pirate door underwater, the pressure at the bottom is larger than the pressure at the top, so the center of pressure is shifted downwards. Similar to the derivation of centroid (in sec- tion 8.3), we are looking for the “balancing point” of all of the forces. With just a little effort, one can show that the appropriate formula- tion to compute the center of pressure is given by The calculation of center of pressure

(COP) for this project is also necessary in other applications, such as wing analysis (in aeronautical design). The complex shape of the wing results in a variable pressure field, and one often needs to understand the point of action associated with that pressure field.

xcp =

∫ b a xP(x)w(x)dx∫ b a P(x)w(x)dx

=

∫ b a xP(x)w(x)dx

FHS . (3)

**P**roject tasks: **Calculus Homework Help**

- Choose an appropriate shape for your door. Feel free to download something from the internet to use as your door shape. Include a figure in your report that gives the shape, where you should make sure that you indicate the appropriate scaling on that figure.
- Use numerical integration to compute the integral required to find the total hydrostatic forces. I suggest using Simpson’s Rule. Recognize that you are not integrating w(x). Rather you are trying to do the integral of (2).
- To find the center of pressure, you will first need to compute the numerator of (3). As above, you will need to use Simpson’s rule to approximate that integral.
- Prepare a technical report describing you analysis.

Use the Math Paper Guide (by Russel Prime) to help you in preparation of your paper. It is posted on Moodle.

#### Additional Instructions:

- Your report must be type written. It should be organized into sections, which are made up of paragraphs, which are composed of complete sentences.
- There is no maximum length. Single spacing is fine.
- You may use whatever choice of bibliographic style you prefer.
- YOU MAY RECEIVE NO HELP FROM ANY LIVE PERSON OTHER THAN YOUR INSTRUCTOR for any of the mathematical work.
- No sources of live help are to be used except as listed above.
- Make sure you include an appropriate ACKNOWLEDGEMENTS section in your paper if you take advantage of any resources.
- You may use any other reference material (your book, the web, etc.) but you must include an appropriate citation and/or reference in your paper.
- EQUATIONS SHOULD BE PROPERLY TYPESET USING THE EQUATION EDITOR CAPABILITY OF YOUR WORD PROCESSOR.
- Please keep in mind: this is a writing assignment, not a homework problem. The grading will primarily be based on your essay and writing, not simply whether the math is correct.

Please consider the following questions of thought which might be appropriate for a discussion section of your paper:

- What are the sources of error in the computation?
- Would the answer get more accurate if you used a finer partition for your Simpson calculation?
- Is there a limit to the accuracy that you might achieve.

- Introduction
- Mathematical and Physics (hydrostatics) background information.
- Project tasks