Problem Set 10—Multiple Integrals and Line Integrals

Purpose

This problem set helps you understand multiple integrals and their applications, and line integrals of scalar-valued functions. When you finish this problem set, you should be able to (1) evaluate multiple integrals over non-rectangular regions, (2) use multiple integrals to evaluate such things as areas, volumes, masses, moments, etc., and (3) evaluate line integrals of scalar functions.

Background

This problem set is based on material in chapter 15 and section 16.1 of our textbook. We discussed multiple integrals (chapter 15) in class between March 30 and April 7. We discussed line integrals on April 12.

Activity

Solve each of the following problems:

Problem 1

Evaluate

Give an interpretation of this integral as a “volume” (or, technically, hypervolume) that helps you make sense of the value you calculated.

Problem 2

Exercise 58 in section 15.2 of our textbook (find the volume of a solid between x² and the section of the xy plane bounded by the curves y = 2 - x² and y = x)

Problem 3

Exercise 18 in the “Additional and Advanced Exercises” section of chapter 15 in our textbook (find the centroid of a “boomerang shaped” region between 2 parabolas). “Centroid” means the center of mass of an object of constant density; assume the density of the boomerang is 1.

Problem 4

Exercise 10 in section 16.1 of our textbook (find the integral of x - y + z - 2 along the segment x = t, y = 1 - t, z = 1 from (0,1,1) to (1,0,1)).

Problem 5

Exercise 28 in section 16.1 of our textbook (find the integral of (x+y²) / √(1+x²) along the curve y = x²/2 from (1,1/2) to (0,0)).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.