1 Answer

Tess · Answer 1 · 2023-03-14T10:47:59+0000

Hello there. To solve this question, we'll have to remember some properties about polar curves and lengths of parameterized curves.

Given the following polar curve defined as:

$r=7\sin(2\theta)$

First, remember that for the polar curve

$r=a\sin(n\theta)$

The number of petals defined by n depends on its parity:

If n is even, we have 2n petals.

If n is odd, we have n petals.

For example, compare

$r=\sin(3\theta)$

The graph is as follows:

And the graph of

$r=\sin(4\theta)$

Its graph is

Okay. So in the case of the polar curve in the question, first note that it has a amplitude of 7 (that is, the distance between the origin and the farthest point from it in a petal is 7)

And it might have 4 petals, hence its graph is:

Notice that the line passing through one petal is the line representing the angle pi/4.

To solve for the arc length, we use the following formula:

$\int_a^b\sqrt{(r(t))^2+\left(\frac{\mathrm{d}(r(t))}{\mathrm{d}t}\right)^2}\mathrm{d}t$

Where a and b are the angles in which the petal is between.

In this case, we'll calculate half of it, integrating from the angle it starts up to pi/4. Hence, doubling the result, we get the final answer.

In this case, we find the intersection of it with the origin making:

$\begin{gathered} r=7\sin(2\theta)=0 \\ \\ \Rightarrow\theta=0 \end{gathered}$

Integrating it up to pi/4, we get:

$\begin{gathered} \int_0^{(\pi)/(4)}2\cdot√((7\sin(2\theta))^2+\left[\left(7\sin(2\theta)\right)'\right]^2)\,\mathrm{d}t \\ \\ \int_0^{(\pi)/(4)}2\cdot√(49\sin^2(2\theta)+196\cos^2(2\theta))\,\mathrm{d}t \end{gathered}$

Notice we can rewrite the radical as:

$\begin{gathered} 49\sin^2(2\theta)+196\cos^2(\theta)=49\sin^2(2\theta)+49\cos^2(2\theta)+147\cos^2(2\theta) \\ \\ \Rightarrow49+147\cos^2(2\theta) \end{gathered}$

Such that we get:

$\int_0^{(\pi)/(4)}2\cdot√(49+147\cos^2(2\theta)^)\,\mathrm{d}t$

Whereas

$\begin{gathered} \cos(2\theta)=2\cos^2(\theta)-1\text{ hence} \\ \\ \cos^2(2\theta)=(2\cos^2(\theta)-1)^2=4\cos^4(\theta)-4\cos^2(\theta)+1 \end{gathered}$

Therefore we get