Part 1 of Green's Theorem and the Planimeter

Green's Theorem and the Planimeter

Part 1: Iterated Integrals and Line Integrals

The first few questions in this part are designed to help you review iterated integrals and, in particular, to review using a computer algebra system (CAS) to evaluate these integrals. Your worksheet contains the commands to evaluate the two integrals described in the following figures.


$\begin{displaymath}\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} sin(x \cdot y) \; dy dx \end{displaymath}$	$\begin{displaymath}\int\!\!\int_{D} \sqrt{x^{2}+y^{2}} \; dA \; \end{displaymath}$ where D is the unit disk centered at the origin.

If you find this material or the computer commands unfamiliar, you might consider reviewing the topic of iterated integrals by working through the modules Double Integrals I and/or Double Integrals II.

Use the command in your worksheet to evaluate the integral $\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} sin(x \cdot y) \; dy dx$ . Does your CAS have any difficulty finding a numerical value for the integral? If so, what do you attribute this to?
Plot the graph of the function $z=f(x,y)=\sqrt{x^{2}+y^{2}}$ , and use the geometry of this graph to evaluate the integral $\int\!\!\int_D \sqrt{x^{2}+y^{2}} \; dA$ where D is the unit disk centered at the origin.
Use your CAS to evaluate the integral $\int_{D}\!\!\int \sqrt{x^{2}+y^{2}} \; dA$ in step 2.
Set up and evaluate the integral:

$\begin{displaymath}\int\!\!\int_G \ln(1+x \cdot y) \; dA \; , \end{displaymath}$

where is the region enclosed by the ellipse

$\begin{displaymath}\frac{x^2}{9} + \frac{y^2}{16} = 1 \; . \end{displaymath}$

Next, you will practice setting up and evaluating line integrals. If a curve C in the plane is parametrized by a function $\gamma: [a,b] \rightarrow$ R $^{2}$ , say $\begin{displaymath}\gamma(t) = (\gamma_{1}(t),\gamma_{2}(t)) \; , \end{displaymath}$ , then

$\begin{displaymath}\int_{C} P(x,y) \; dx + Q(x,y) \; dy=\int_a^b P(\gamma_1(t),\... ...\gamma_1(t),\gamma_2(t)) \cdot \gamma_2^{\prime}(t) \; dt \; . \end{displaymath}$

Use your CAS to evaluate the line integral $\int_C P(x,y) \; dx + Q(x,y) \; dy$ when , , and C is the boundary of the square with vertices at (0,0), (1,0), (1,1), and (0,1), traced out in a counter-clockwise direction.
How can you express the line integral in step 5 as a sum of four simple, one-dimensional definite integrals? How can this be used to explain the result of your calculation above?

Set up and evaluate the integral $\int_{C} P(x,y) \; dx + Q(x,y) \; dy$ when $P(x,y) = x \cdot y$ , , and C is the boundary of the circle (x - 1)² + (y - 1)²= 1, traced out in a counter-clockwise direction.

Next, you will set up and evaluate some iterated integrals and some line integrals from scratch. Note that you can save a lot of typing in your worksheet by copying, pasting, and editing.

Set up and evaluate the following line integrals, where C is the square with vertices at (0,0), (1,0), (1,1) and (0,1), traced out in a counter-clockwise direction.

$\begin{displaymath}\int_{C} (x \cdot y)^{2} \; dx \end{displaymath}$
$\begin{displaymath}\int_{C} sin(x \cdot y) \; dy \end{displaymath}$
$\begin{displaymath}\int_{C} (x \cdot y)^{2} \; dx + sin(x \cdot y) \; dy \end{displaymath}$

Set up and evaluate the following iterated integrals, where D is the interior of the square in step 8.
- $\begin{displaymath}\int\!\!\int_D y\cdot \cos(x \cdot y) \; dA \end{displaymath}$
- $\begin{displaymath}\int\!\!\int_D - 2\cdot (x \cdot y)\cdot y \; dA \end{displaymath}$
- $\begin{displaymath}\int\!\!\int_D (y\cdot \cos(x \cdot y) - 2\cdot (x \cdot y)\cdot y) \; dA \end{displaymath}$
Set up and evaluate the following line integrals, where C is the circle (x - 1)²+ (y - 1)²= 1, traced out in a counter-clockwise direction.
- $\begin{displaymath}\int_{C} (x \cdot y)^{2} \; dx \end{displaymath}$
- $\begin{displaymath}\int_{C} sin(x \cdot y) \; dy \end{displaymath}$
- $\begin{displaymath}\int_{C} (x \cdot y)^{2} \; dx + sin(x \cdot y) \; dy \end{displaymath}$
Set up and evaluate the following iterated integrals, where D is the interior of the circle in step 10.
- $\begin{displaymath}\int\!\!\int_D y\cdot \cos(x \cdot y) \; dA \end{displaymath}$
- $\begin{displaymath}\int\!\!\int_D - 2\cdot (x \cdot y)\cdot y \; dA \end{displaymath}$
- $\begin{displaymath}\int\!\!\int_D (y\cdot \cos(x \cdot y) - 2\cdot (x \cdot y)\cdot y) \; dA \end{displaymath}$
What relationships have you noticed between the values of the iterated integrals and the values of the line integrals? Summarize the patterns that you have noticed as a conjecture about how the values of line integrals and iterated integrals may be related.

modules at math.duke.edu