Sketching a polar curve

Continued off the question I asked earlier, I also have to sketch the curve.

$r^2=−4\sin(2\theta)$

So I have to set up a table of values I'm assuming. How do I know what values to choose for $\theta$. Then I'm just lost all together trying to plot the theta and $r$ values to get the curve.

Oh and one quick other question. If it also asks me to find all the points of intersections and to set up the integral to evaluate the area inside after sketching the curve, what does that mean? Do I have to pick a top and bottom or something? Confused about that.

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2 Answers

As a rule, it helps to know how often your function repeats itself (if ever). It is usually found by looking at the co-efficient of the θ (in this case 1). You then smoothly connect where it is at 0 to 2π in 4*co-efficient number of even increments. Remember that if r is negative, its reflected over the x axis. Generally also a good idea to have the zeros of the function, but not always necessary.

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To get started, you might try values of $\theta$ where it's easy to calculate stuff, like $\theta=0,\pi/12,\pi/4,\pi/2,\dots$. Maybe even better to note first that since $r^2\ge0$ it only makes sense to take $\theta$ with $\sin2\theta\le0$.

You won't get very far with the integral until you have a good sketch of the curve.

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