Friday, December 9, 2016

equation sheet for exam four

Line and surface integrals are complicated; use the equation sheet for exam four as you work on the homework and discussion problems.

Friday, December 2, 2016

surface integrals (part 1), section 14.6

Today we'll work on integrating scalar functions over common surfaces. To save time I'll let you read about the vector and scalar surface area elements on your own. This table contains some common surface area elements:

Wednesday, November 30, 2016

lecture 37: green's theorem

We'll look at another version of the fundamental theorem (called Green's Theorem, section 14.4) that connects double integrals to line integrals over closed curves in $\mathbb{R}^2$. 

Tuesday, November 29, 2016

exam three results

On the Thursday before the Thanksgiving break, 218 brave calcunauts took exam three. Another 5 or 6 will take a makeup exam. You should see your score in WyoCourses now; if not, let me know.

The mean is 75.5 and the quartile scores are 67, 75.5 and 84.25. One brave calcunaut has a perfect score. For comparison, last semester the mean was 73.4 and the quartile scores were 66, 74.5, and 83.

Monday, November 28, 2016

lecture 36: conservative vector functions

We will use the fundamental theorem from calculus I to construct a fundamental theorem for line integrals involving $\vec{F} \cdot d\vec{r}$ (14.3). The fundamental theorem only works if the vector function is conservative: $\vec{F} = \vec{\nabla}f$. We'll create a test to tell whether a generic vector function is conservative, and we'll figure out how to construct the potential function $f(x,y,z)$ for a conservative vector function.

Saturday, November 26, 2016

need a job next semester?

Trigonometry needs an SI leader. If you are interested, contact Nathan Clements at

Nathan provided this job description: SI leaders organize weekly SI sessions where they help Trigonometry students, as a peer, to learn in collaboratively.  They help answer their questions and help teach them quality study skills. SI Leaders work 7-8 hours a week, with most of those hours spent getting training and staying up to date with the material in Trigonometry.

Tuesday, November 22, 2016

no office hours today

I am in hiding today (11/22/2016), trying to finish my part of the exam grading. Enjoy the break.

no discussion section today

Due to Thursday being a holiday there are no discussion sections today (11/22/2016).

Monday, November 21, 2016


What are my chances of getting my exam back Tuesday? If you mean Tuesday, Nov. 22, the chances are zero. If you mean Tuesday, November 29, the chances are 99%.

lecture 35: line integrals and the fundamental theorem

I'll review the line integrals from last week and talk about what happens when we reverse the direction of integration. Then, we'll look at alternate ways to express work and flux integrals (14.2) and compute some work integrals using conservative force fields (14.3).

Sunday, November 20, 2016

if only we could all write this well, part 3

Grading is going slowly, but I've seen lots of excellent solutions to problems 3 and 4. Here are two examples I came across today.


click to embiggen

Friday, November 18, 2016

lecture 34: line integrals (work and flux)

We will continue working with line integrals (14.2), replacing $f(x,y,z)$ with components of a vector function $\vec{F}(x,y,z)$. One application is to compute the work done by $\vec{F}$ on a mass that moves along a curve: $$\textrm{work = }\int \vec{F} \cdot \hat{T} \, ds $$ where $\hat{T}$ is a unit tangent vector to the curve. Another application is to compute the flux of $\vec{F}$ across a curve in $\mathbb{R}^2$: $$\textrm{flux = }\int \vec{F} \cdot \hat{n} \, ds $$ where $\hat{n}$ is a unit normal vector to the curve. To evaluate these line integrals we'll use the parametric equations for the curve to make a change of variables (like last time).

Thursday, November 17, 2016

exam 3 solutions

I will add details to these solutions in the next few days. Please let me know if you find any mistakes.
  1. Dean McClure pointed out that, in problem 3, I forgot to update the limits of integration after making the substitution $u=y^2$. He is correct and that mistake is fixed as of 6:20 am (10/18).

exam three approaches

Exam three is history! The exam covers sections 13.1 through 13.7No electronic devices are allowed at the exam.

We will provide you with this equation sheet. Be careful, some facts are not on the equation sheet. You need to know how to find limits of integration and integrate. You need to know how to calculate an area using a double integral and how to calculate a volume using a double or triple integral. You should also know the cosine and sine of common angles like $0$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$, $\pi$, and $2\pi$ radians.
Your exam room is a function of the first three letters of your last name.

  • Aaa through Iri, go to CR 302
  • Jag through Rac, go to CR 306
  • Rai through Zzz, go to CR 310

  • We are again sharing the rooms with Calculus II students. Make sure you are not sitting next to another Calculus III student.

    To practice for the exam, use the problems from MyMathLab, your discussion section, and the mock exams (you'll need to look at mock exam 3 for problems from sections 3.1 through 13.6, and mock exam 4 for problems from 13.7) and examples from the text. If you don't understand something ask questions at your discussion section and during our office hours.

    An engineering professor talks about learning:

    Wednesday, November 16, 2016

    lecture 33

    By popular demand we used the lecture to review for exam 3.

    Monday, November 14, 2016

    lecture 32: line integrals

    I'll talk about how to integrate a scalar function $f(x,y,z)$ along a curve in $\mathbb{R}^3$ using the scalar arc length element $ds=| \, d\vec{r} \, |$. The trick (14.2) is to make a change of variables using the parametric equations of the curve: $$\int f(x,y,z) \, ds = \int f(t) \, | \, d\vec{r}^{\, \prime}(t) \, | \, dt. $$ This type of integral is used to compute physical properties of wires (think length, mass, and center of mass).

    We'll look at section 14.1 after we talk about a second type of line integral on Wednesday.

    Friday, November 11, 2016

    inverse substitution

    lecture 31: change of variables

    We'll wrap up any undone mass integrals and rewrite several double integrals with complicated regions of integration by applying either a forward or inverse substitution (13.7).

    Wednesday, November 9, 2016

    lecture 30: mass integrals

    We'll continue to work on triple integrals in spherical and cylindrical coordinates (13.5), but we'll use them to calculate masses and centers of mass for three dimensional regions (13.6), instead of volumes.