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Chapter 8: Problem 76

Sketch a graph of the polar equation. $$ r^{2}=4 \sin \theta $$

Short Answer

Expert verified
The graph of the polar equation \(r^{2}=4 \sin \theta\) is a semicircle with radius 2 resting against the positive side of the y-axis in the polar coordinate system.

Step by step solution

01

Simplify the Equation

The equation given in the problem is in polar form. For easier manipulation, it鈥檚 helpful to simplify it into a more recognizable form. One way to do so is by taking the square root of each side. This results in the following equation: \( r = 2 \sqrt{\sin \theta} \).
02

Identify the Graph's Features

The equation \( r = 2 \sqrt{\sin \theta} \) is recognizable as a semicircle in a rectangular coordinate system with a radius of 2. However, it should be noted that when \(\sin \theta\) is negative, \(\sqrt{\sin \theta}\) is undefined. Therefore, this graph will only exist when \(0 \leq \theta \leq \pi\). It's centered at the pole with the bottom touching the pole.
03

Sketch the Graph

The graph will appear as a semicircle against the positive side of the y-axis, with the pole serving as the center. Starting from \(\theta = 0\), the circle will start from the pole and then, increase in radius until \(\theta = \frac{\pi}{2}\). After that, the radius will start to decrease again until \(\theta = \pi\), at which point the radius will be zero again. The sketch of the graph should reflect these properties.

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Most popular questions from this chapter

Show that the polar equation for \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta} \cdot \quad\) Ellipse

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t-1, \quad y=\frac{t}{t-1} $$

In Exercises 47 and 48, use the integration capabilities of a graphing utility to approximate to two decimal places the area of the surface formed by revolving the curve about the polar axis. $$ r=4 \cos 2 \theta, \quad 0 \leq \theta \leq \frac{\pi}{4} $$

Use the results of Exercises \(31-34\) to find a set of parametric equations for the line or conic. $$ \text { Ellipse: vertices: }(\pm 5,0) ; \text { foci: }(\pm 4,0) $$

Describe what happens to the distance between the directrix and the center of an ellipse if the foci remain fixed and \(e\) approaches 0 .
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