Wednesday, November 25, 2015

exam three results

If you took this exam you may view your score and current total score in WebAssign by clicking on the Grades link.

We graded 201 exams this time; another five or six people will take a makeup exam after the break. The mean is 74.5 and the quartile scores are 63, 76, and 87.5. Slightly more than 50% of the group scored 76 or higher and slightly more than 25% scored 87.5 or higher. Last semester the quartile scores were 68.25, 78, and 85.75.

The median scores on the individual problems are:

 16/20 10/10 9/10 7/10 4/6 12/12 8/12 10/10 8/10

You can't see it in the median scores, but too many people in the bottom two quartiles hurt their scores by not graphing the regions of integration (problems 3, 7 and 8). It's hard to get the limits right if you can't see what you are doing.

Once again, many people produced beautiful, detailed solutions to these problems. I'll post examples soon.

If you scored 65 or below (29% of the group), or you just want an A in the course, give some thought to these lists of good and bad study habits: ten things to do and ten things not to do.

Monday, November 23, 2015

green's theorem, section 16.4

We'll ponder our results from Friday and then move on to discuss Green's Theorem, a new extension of the fundamental theorem of calculus. Green's theorem relates a double integral over a region in the plane to a line integral around the boundary of the region. We'll verify Green's theorem and then use the theorem to perform some neat tricks.

Friday, November 20, 2015

conservative vector functions, section 16.3

We'll will finish Wednesday's project and see why a vector function of the form $\vec{F}(x,y,z)=\vec{\nabla}f(x,y,z)$ is said to be conservative. The short story is that the fundamental theorem of calculus requires that energy (kinetic plus potential) be conserved if the force is conservative. And, this last statement explains why math rulz!

If there is time, we'll start work on the next version of the fundamental theorem, Green's Theorem in section 16.4. Green's Theorem links double integrals to line integrals.

Thursday, November 19, 2015

exam three solutions

Please let me know if you find mistakes or unclear explanations in my solutions to exam three. I fixed the typo in problem 6c. On the middle integral, the upper limit should have been $z=2$.

exam 3 is history!

Exam three starts at 5:15 pm this afternoon (Thursday, November 19). The exam covers sections 15.1-15.5 and 15.7-15.10.

Calculators and computers are not allowed at the exam. We will provide you with an equation sheet. You can find the equation sheet on the last page of each mock exam. Practice your integration; there are some complicated integrals on the mock exams.

Your exam room is a function of your discussion section. If possible, leave an empty seat between you and the next calcunaut.

• Section 20, Business Auditorium
• Section 21, Business Auditorium
• Section 22, History 57
• Section 23, Geology 216
• Section 24, Geology 216
• Section 25, History 57

• To practice for the exam, use your WebAssign, discussion, and mock exam problems. Solutions are available for these problems. If you don't understand something ask questions at your discussion section and during our office hours.

Wednesday, November 18, 2015

the fundamental theorem, section 16.3

We will use the fundamental theorem from calculus I to construct a fundamental theorem for line integrals involving $\vec{F} \cdot d\vec{r}$. 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.

Monday, November 16, 2015

line integrals, section 16.2

On Friday we integrated the product $f(x,y,z) \, ds$, where $ds = | d\vec{r}|$, along a space curve. We were able to solve the problem by using the parametric equations for the curve to make a change of variables so that $f(x,y,z) \, ds = f(t) | \vec{r}^\prime(t) | dt$. With this integral we can compute the length of a curve, and the mass, center of mass, and moment of inertia of a wire.

Today we'll integrate another product, $\vec{F}(x,y,z) \cdot d\vec{r}$ along several space curves. Again, we'll use the parametric equations for the curve to make a change of variables so that we can make quick work of the integrals. With this integral we can compute the change in kinetic energy of a mass moving through space under the influence of a force field, or the voltage in a circuit due to an external electric field.

Friday, November 13, 2015

line integrals, section 16.2

Today I'll show how to integrate a function $f(x,y,z)$ along a curve in $\mathbb{R}^3$ using the scalar arc length element $ds=\sqrt{(dx)^2+(dy)^2+(dz)^2}$. The trick is to use the parametric equations of the curve to make a change of variables. This type of integral is used to compute physical properties of wires (think mass, center of mass, and moment of inertia).

On Monday we'll look at a second type of line integral where we integrate a component of a vector field along a curve. To ready ourselves we'll look at a few examples of vector fields in $\mathbb{R}^2$ (section 16.1).

 Psychedelic!

Wednesday, November 11, 2015

general transformations, section 15.10

We'll solve several double integrals with complex regions of integration by applying either a forward or inverse substitution. If you are reading along, the early part of section 15.10 is needlessly scary; a safe place to start is the redlined box on page 1044. This video shows another example of an inverse substitution.

Tuesday, November 10, 2015

discussion, week 11

For this week's quiz you'll need to know how to convert Cartesian equations to cylindrical coordinates and how to write limits of integration in cylindrical and Cartesian coordinates.

The discussion problems deal with triple integrals, volume calculations, and general transformations.

Monday, November 9, 2015

triple integrals, section 15.9

I'll review spherical coordinates and then set up some integrals in spherical coordinates.

Friday, November 6, 2015

triple integrals, section 15.8

We'll set up triple integrals in cylindrical coordinates and talk about the spherical coordinate system (15.9).

Wednesday, November 4, 2015

triple integrals, sections 15.7 and 15.8

I'll finish the problem I was working on Monday by showing how to project a space curve on to a coordinate plane. We'll do this using parametric equations and also Cartesian equations.

Then we'll turn our attention to regions of integration that have cylindrical symmetry, symmetry about one axis. We'll use a mash up of polar coordinates and the $z$-axis to make quick work of these cases.

Tuesday, November 3, 2015

we should all be writing solutions like these

People produced some really sweet solutions to exam two. Here are two examples.

Click to embiggen.

Monday, November 2, 2015

discussion, week 10

This week's problems are about setting limits of integration for triple integrals. All the problems use Cartesian coordinates and several require projections to figure out the limits of integration for the outside integrals.

This week's quiz is about double integrals in Cartesian coordinates. You'll have to evaluate a simple double integral, and you'll need to know how to set up and reverse limits of integration.

triple integrals, section 15.7

We'll look at several, slightly, complicated regions of integration today, ones where we can use our projection and parametrization skills. We are sticking with Cartesian coordinates.

Friday, October 30, 2015

triple integrals, section 15.7

We are skipping over section 15.6 but don't worry. You'll have a chance to compute a few surface areas in the second half of chapter 16, after we figure out how to parametrize surfaces.

Today we'll start on triple integrals in Cartesian or rectangular coordinates (15.7). We'll set up integrals that give the volume or moment of inertia of various solid regions, and we'll review the process of determining limits of integration for non-rectangular regions.

Wednesday, October 28, 2015

more applications, section 15.5

We'll continue to compute interesting physical quantities with double integrals. So far we've computed volumes, areas, masses, and first moments of mass distributions. Today we'll add second moments of the mass distribution, also known as moments of inertia.