Problem Set 11 — More about Integrals

Purpose

This exercise reinforces your understanding of definite integrals of multivariable functions and of scalar line integrals. It contributes to the following learning outcomes for this course:

• Outcome 7. Evaluate double and triple integrals
• Outcome 11. Evaluate line integrals directly and by the fundamental theorem.

Background

This exercise is based on material in sections 4.3 and 5.2 in our textbook. We covered those sections in classes between November 7 and 14.

Activity

Solve each of the following problems.

Problem 1

Exercise 18 in section 4.3E of the textbook:

Integrate $f(x,y)=x^2+xy$ over the polar region defined by $1\le r\le 2$ and $\pi \le \mathrm{\Theta }\le 2\pi$.

Problem 2

Exercise 26 in section 4.3E of the textbook:

Show that the following equation holds between an integral in rectangular coordinates and a (supposedly) equivalent one in polar coordinates. Then evaluate the integral in whichever coordinate system you think will be easiest.

Problem 3

Find the value of

where $C$ is the right half of the circle $x^2+y^2=16$.

Problem 4

Find the value of

where $C$ is the 2-part path that follows a straight line from point $(0,0,0)$ to point $(1,0,0)$, and then from there to point $(1,1,1)$.

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. 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.