Math 223 Multivariable and Line Integrals

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 11. Evaluate line integrals directly and by the fundamental theorem
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 based on material in sections 4.3, 5.1, and the first part of 5.2 in our textbook. We discussed section 4.3 in class on April 17, section 5.2 on the 18th and 19th, and 5.1 on the 21st.

This exercise also asks you to plot vector fields with Mathematica, which we discussed in class on April 21.

Activity

Solve each of the following problems.

Problem 1

Exercise 26 in section 4.3E of our textbook.

Show that the following claim is true by explaining how the rectangular integral on the left converts into the polar integral on the right (or vice versa). Then evaluate the integral in either rectangular or polar form, whichever you think will be easiest.

\int_{1}^{2} \int_{0}^{x} x^{2} + y^{2} d y d x = \int_{0}^{\frac{π}{4}} \int_{\sec θ}^{2 \sec θ} r^{3} d r d θ

Problem 2

The double integral of a function $f$ over a region $D$ in the $x y$ plane is informally described as the volume between the region $D$ and the graph of $f$ . We saw an example of this in class on April 17, when we integrated a hemisphere of radius 1 over its circular base and got the same numeric answer that the geometric formula for the volume of a hemisphere gives.

Repeat this exercise, but this time let the radius of the hemisphere be an unknown value $R$ , and show that the integral gives the general formula for the volume of a hemisphere:

V = \frac{2}{3} π R^{3}

Hints: to do this exercise, you’ll need to find an equation for a hemisphere of radius $R$ centered at the origin, then find a way to describe the base of that hemisphere in the $x y$ plane, and finally integrate your hemisphere formula over that base region.

Problem 3

Find the value of

\int_{C} x y^{4} d s

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

Problem 4

For each of the following vector fields, give the value of the vector field at the origin, and then use Mathematica to plot the field in a small region centered on the origin (you can decide for yourself what “small” should mean here).

Vector Field A

\vec{F} (x, y) = ⟨ y - 1, x + 1 ⟩

Vector Field B

\vec{G} (x, y, z) = ⟨ x, y, z ⟩

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.