Question

Sketch the following regions \(R\). Then express \(\iint_{R} f(r, \theta) d A\) as an iterated integral over \(R.\) The region inside the lobe of the lemniscate \(r^{2}=2 \sin 2 \theta\) in the first quadrant

Short answer

Question: Sketch the curve represented by the polar equation \(r^2 = 2\sin 2\theta\) in the first quadrant, and express the double integral of a function f(r, θ) over the region R inside the lobe of the curve as an iterated integral. Answer: The double integral of f(r, θ) over the region inside the lobe of the curve \(r^2 = 2\sin2\theta\) can be expressed as an iterated integral as follows: \(\iint_{R} f(r, θ) dA = \int_{0}^{\frac{\pi}{4}}\int_{0}^{\sqrt{2\sin 2\theta}} f(r, θ) r dr d\theta\)

Step-by-step solution

Sketch the polar equation

To sketch the curve \(r^2 = 2\sin2\theta\), first, we need to observe the behavior of the curve at different values of θ. Let us create a table of values for r and θ: $\begin{array}{c|c} \theta & r^2 \\ \hline 0 & 0 \\ \frac{\pi}{12} & 1 \\ \frac{\pi}{6} & 1 \\ \frac{\pi}{4} & 0 \\ \frac{\pi}{3} & -1 \\ \end{array}$ Plotting these points and drawing a curve, we can see that this curve is a lemniscate with a single lobe in the first quadrant.

Find the limits of integration for r and θ

Now we need to express the region R inside the lobe of the lemniscate in the first quadrant as an iterated integral. First, we need to find the ranges for r and θ. By looking at the graph, we can see that θ varies from 0 to \(\frac{\pi}{4}\) (first quadrant), so: \(0 \leq \theta \leq \frac{\pi}{4}\) As for r, it varies from 0 to the lemniscate curve: \(r(\theta) = \sqrt{2\sin 2\theta}\). So, the limits for r are: \(0 \leq r \leq \sqrt{2\sin 2\theta}\)

Convert the double integral in Cartesian coordinates to polar coordinates

To convert the double integral \(\iint_{R} f(r, \theta) d A\) from Cartesian to polar coordinates, we need to replace \(dA\) with \(r dr d\theta\). This substitution is necessary because the area element in polar coordinates is not a simple square, but a sector of a circle that has a small width \(dr\) and a small angle \(d\theta\): \(dA = r dr d\theta\)

Write the final expression for the iterated integral

Now, we can express the double integral as the following iterated integral using the limits of integration for r and θ that we established earlier: \(\iint_{R} f(r, \theta) d A = \int_{0}^{\frac{\pi}{4}}\int_{0}^{\sqrt{2\sin 2\theta}} f(r, \theta) r dr d\theta\) This is the final expression for the double integral of f(r, θ) over the region R inside the lobe of the lemniscate.

Key concepts

Polar Coordinates

Polar coordinates provide a unique way to describe the position of a point in a plane using two values: the radial distance from a fixed point (the origin) and the angle from a fixed direction (usually the positive x-axis). Unlike Cartesian coordinates, which use an x and y coordinate to describe a point, polar coordinates are ideal for dealing with situations involving circles or other round shapes.

With polar coordinates, we denote each point as \(r, \theta\), where \(r\) is the radial distance and \(\theta\) is the angle in radians.

• \(r\) is the distance from the origin to the point.
• \(\theta\) is the angle measured in radians from the positive x-axis.

When converting a double integral from Cartesian coordinates to polar coordinates, the differential area element \(dA\) changes to \(r \, dr \, d\theta\). This is crucial in ensuring the integral respects the sector parts of circles involved in polar transformations.

Lemniscate

A lemniscate is a fascinating geometric figure that resembles the shape of a figure-eight or an infinity symbol. Specifically, the equation \(r^2 = 2 \sin 2\theta\) describes a lemniscate in polar coordinates. Understanding and sketching this curve involves knowing how the radius \(r\) changes with respect to the angle \(\theta\).

• The curve typically loops back onto itself, creating two lobes.
• In our example, \(r^2 = 2 \sin 2\theta\), we only focus on the first quadrant lobe, informed by limits such as \(0 \leq \theta \leq \frac{\pi}{4}\).

To sketch this correctly, observe how \(r\) responds to changes in \(\theta\), particularly checking the maximum and minimum points, which help define the extent of the lemniscate. Such understanding helps accurately set up the limits of integration for analyses like finding areas using integrals.

Iterated Integral

The concept of an iterated integral is pivotal in multivariable calculus, particularly for evaluating double integrals. An iterated integral is essentially a way of expressing a multiple integral as a sequence of single integrals.When we deal with polar coordinates, the iterated integral takes the form \(\int_{a}^{b} \int_{c}^{d} f(r, \theta) r \, dr \, d\theta\), with \(r \, dr \, d\theta\) replacing \(dA\) as described previously. This is due to the nature of areas in polar coordinates:

• You first integrate with respect to \(r\) over the interval \([0, \sqrt{2 \sin 2\theta}]\).
• You then integrate with respect to \(\theta\) from 0 to \(\frac{\pi}{4}\).

This setup allows us to handle variables independently, first resolving the radial part and then the angular component. Mastery of iterated integrals enables deeper insights into functions that are defined over rotationally symmetric regions, streamlining calculations and enhancing comprehension.