Name ______Date ______Period ______

AP Calculus AB Project

Volumes of Solids with Known Cross-Section

You and your group (2-4 people) will make a physical model of a solid with known cross-sections perpendicular to the x-axis or the y-axis. The base of your solid is bounded by the graphs of equations determined by your group.

The following guidelines apply:

1) The cross section can be any shape but only two groups may pick the same shape. Popular shapes include squares, oblongs, semi-circles, and triangles. Please tell Ms. Harding what shape you are using for your cross sections. You do not have to use one of the “popular” shapes, just let Ms. Harding know in advance (She will also be very impressed!).

2) Your model must be at least 6 inches in length, width, and height. Your model must have at least 40 cross sections. Your model must be mounted on a sturdy surface (not wood). Cardboard or Styrofoam are excellent materials to use as cross sections and for mounting.

Your presentation must have the following information:

1) A detailed description of the functions used as the base of your solid.

2) An explanation of what each cross section looks like as well as how you determine the area

of each cross section.

3) The volume of the solid as defined by a definite integral. If your problem is not

integrable using the basic rules of integration, you may use the “numerical integration” feature of your graphing calculator. (She will be very impressed if you integrate manually to find the exact volume of your solid.)

Scoring Guidelines:

1) 3-Dimensional Cross-Section Model

- 10 points - Identify the base functions of your solid and the interval.

- 10 points - Identify the function used to calculate the area of each cross section

2) Presentation of the information and calculations

- 10 points – Typed (neatly written) volume unique problem with known cross sections

- 10 points - Use of proper calculus notation

- 10 points - Accuracy of calculations

3) Neatness and appeal of your model and solution

- 15 points – Cross-sections are secured to your base

- 15 points - Aesthetically pleasing

Projects that impress Ms. Harding and show extra effort and performance

may earn 5 extra credit points!

Group Name ______

Member ______

Member ______

Member ______

Member ______

Base Function(s) ______

______

Interval ______

Shape of each Cross-section ______

Perpendicular to the x-axis or y-axis? ______

Area of each cross-section ______

Definite Integral to find the volume of the solid ______

Will you manually integrate to find the volume or use your calculator? ______

Sketch the base of your solid below