Problem Set 10—Multiple and Line Integrals

Purpose

This problem set reinforces your understanding of, and ability to work with, multidimensional integrals. It also begins to develop your understanding of integrals over curves in multiple dimensions.

Background

This exercise is based on material in sections 15.5, 15.6, and 16.1 of our textbook. We will discuss this material in class between November 13 and November 18.

Activity

Solve each of the following problems:

Problem 1

Exercises 22a and c in section 15.5 of our textbook (rewrite the integral from 0 to 1 of the integral from -1 to 0 of the integral from 0 to y² evaluated in the order dz dy dx to be evaluated in the orders (a) dy dz dx, and (c) dx dy dz).

Problem 2

Exercise 33 in section 15.5 of our textbook (find the volume in the first octant of the standard coordinate system of the region between the planes x + y + 2z = 2 and 2x + 2y + z = 4).

Problem 3

Evaluate

Give a geometric interpretation of this integral that helps you make sense of the value you calculated.

Problem 4

Exercise 24 in section 16.1 of our textbook (find the line integral of (√y) / x along ⟨ t³, t⁴ ⟩).

Problem 5

Exercise 32 in section 16.1 of our textbook (find the area of one side of a curved wall whose bottom follows the line 2x + 3y = 6, 0 ≤ x ≤ 6 and whose top lies on the surface f(x,y) = 4 + 3x + 2y).

Problem 6

Exercise 29a in section 15.6 of our textbook (find the mass of a solid in the first octant of the standard coordinate system below the plane x + y + z = 2, with density δ(x,y,z) = 2x).

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.