4.6 Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates

Introduction

Integration in Polar Coordinates

This Preview Activity asks you to practice converting some equations and regions between rectangular and polar coordinates.

For each of the following polar equations, draw a plot in the plane of the curve. Write a sentence for each graph to explain why the equation is satisfied. -$r=1$-$r=3$-$r=0$-$\theta =1$-$\theta = \frac{3\pi}{4}$

Let $f(x,y) = e^{x^2+y^2}$ on the disk $D = \{(x,y) : x^2 + y^2 \leq 1\}$. Our goal in this example is to evaluate $\displaystyle \iint_D f(x,y) \, dA$.

In rectangular coordinates the double integral $\iint_D f(x,y) \, dA$ can be written as the iterated integral

, where the vertically simple slicing is shown in Figure 4.6.1

We cannot evaluate this iterated integral because $e^{x^2 + y^2}$ does not have an elementary antiderivative with respect to either $x$ or $y$. However, since $r^2=x^2+y^2$ and the region $D$ is circular, it is natural to wonder whether converting to polar coordinates will allow us to evaluate the new integral. To do so, we replace $x$ with $r \cos(\theta)$, $y$ with $r \sin(\theta)$, and $dy \, dx$ with $r \, dr \, d\theta$ to obtain

.

The disc $D$ is described in polar coordinates by the inequalities $0 \leq r \leq 1$ and $0 \leq \theta \leq 2\pi$. Therefore, it follows that

. We can evaluate the resulting iterated polar integral as follows:

.

Work through this exercise and explain your reasoning step by step.

We will find the volume by computing a double integral $\iint_R f(x,y)\, dA$ where $R$ is an appropriate region in the $xy$-plane. Sketch the region $R$.

Write inequalities to describe $R$ in polar coordinates. Write a couple of sentences explaining why it would be very difficult to describe $R$ using rectangular coordinates.

Use an interated integral in polar coordinates to find the volume of $V$.

Consider the iterated integral $I = \int_{-3}^{0} \int_{-\sqrt{9-y^2}}^{0} \frac{y}{x^2 + y^2+1} \, dx \, dy.$- Sketch (and label) the region of integration. - Convert the given iterated integral to one in polar coordinates. - Evaluate the iterated integral in (b). - State one possible interpretation of the value you found in (c).

Let $D$ be the region that lies inside the unit circle in the plane. - Set up and evaluate an iterated integral in polar coordinates whose value is the area of $D$. - Determine the exact average value of $f(x,y) = y$ over the upper half of $D$. - Find the exact center of mass of the lamina over the portion of $D$ that lies in the first quadrant and has its mass density distribution given by $\delta(x,y) = 1$. (Before making any calculations, where do you expect the center of mass to lie? Why?) - Find the exact volume of the solid that lies under the surface $z = 8-x^2-y^2$ and over the unit disk, $D$.

For each of the following iterated integrals, - sketch and label the region of integration, - convert the integral to the other coordinate system (if given in polar, to rectangular; if given in rectangular, to polar), and - choose one of the two iterated integrals to evaluate exactly. -$\int_{\pi}^{3\pi/2} \int_{0}^{3} r^3 \, dr \, d\theta$-$\int_{0}^{2} \int_{-\sqrt{1-(x-1)^2}}^{\sqrt{1-(x-1)^2}} \sqrt{x^2 + y^2} \, dy \, dx$-$\int_0^{\pi/2} \int_0^{\sin(\theta)} r \sqrt{1-r^2} \, dr \, d\theta.$-$\int_0^{\sqrt{2}/2} \int_y^{\sqrt{1-y^2}} \cos(x^2 + y^2) \, dx \, dy.$