Purpose
This exercise develops your understanding of integration in multiple dimensions. It therefore contributes to the following learning outcomes for this course:
- Outcome 7. Evaluate double and triple integrals
- Outcome 8. Use multiple integrals to find area and volume
- Outcome 12. Use technological tools such as computer algebra systems or graphing calculators for visualization and calculation of multivariable calculus concepts.
Background
This exercise is mainly based on material in sections 4.1 through 4.4 of our textbook. We covered that material in classes between April 7 and 14. This problem set also asks you to evaluate multiple integrals with Mathematica. We saw some examples of that in class on April 14.
Activity
Solve each of the following problems.
Problem 1
Let region
Part A
Evaluate
over region
Part B
Use a double integral to find the area of region
Problem 2
One of the properties of double integrals that our textbook states is that over a
rectangular region
Justify this claim by showing how to express the double integral from the left side of the equation as an iterated integral and then rearrange it into the product on the right side.
Problem 3
Evaluate
Give an interpretation of this integral as a “volume” (or, technically, hypervolume) that helps you make sense of the value you calculated.
Problem 4
(Based on Exercise 11 in section 4.2E of our textbook.)
Evaluate
Where
Check your result by also evaluating the integral with Mathematica.
(Hint: this is an ugly-looking integral that turns out to have a simple result if you simplify it right.)
Follow-Up
I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above, and should ordinarily last half an hour. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.