Chapter 6, Problem 72RE

Solution

To plot: The graph of the polar equation $r^2=2\sin 2\theta ,0\le \theta \le 2\pi$ and analyze the graph.

The graph of the polar equation $r^2=2\sin 2\theta ,0\le \theta \le 2\pi$ as shown in Figure (1) is bounded and symmetric about origin.

Given information:

The polar equation is $r^2=2\sin 2\theta ,0\le \theta \le 2\pi$ .

Calculation:

Plot the graph of the given polar equation.

  

Figure (1)

From the graph, it can be analyzed that the graph is bounded. The domain and range of the graph are $\left[0,\frac{\pi }{2}\right]{\displaystyle \cup \left[\pi ,\frac{3\pi }{2}\right]}$ and $\left[0,\sqrt{2}\right]$ . It is symmetric about origin. The maximum value of $r$ is $\sqrt{2}$ . It has no asymptotes.

Therefore, the graph of polar equation $r^2=2\sin 2\theta ,0\le \theta \le 2\pi$ is bounded and symmetric about origin.