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I would like to have a function that gives the area of a rectangle at a certain theta of the spiral r=theta. The height of the rectangle is the y value of the point on the spiral and the base of the rectangle is the x value of the point on the spiral. I know the area would then just be xy (positive if its above the x axis and negative if below the x axis) for each point (x,y) on the curve. I know that r=theta can be written in cartesian coordinates as tan(sqrt(x^2+y^2)) = y/x, but I don't know if I can solve this equation for x or y.

For example, given F(5 * sqrt(2)) = 25:

blue square, green spiral on graph

The red line is the the angle on the spiral, and the blue lines are the high and base of the rectangle.

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    $\begingroup$ Although this is an interesting question, it is not research level, so, despite @JonathanLove's nice answer and comment, it will probably be closed. For future questions at this level, try our sister site MSE. $\endgroup$
    – LSpice
    Commented Oct 9, 2021 at 1:40

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A parametric form for this spiral is given by $x=\theta\cos\theta$ and $y=\theta\sin\theta$. So the area of the rectangle is $\theta^2\sin\theta\cos\theta=\frac{\theta^2}{2}\sin(2\theta)$.

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    $\begingroup$ Also worth pointing out: the example you shared is not quite right. The correct value is $F(5\sqrt{2})=24.9996914\ldots$. The reason your picture looks the way it does is because $5\sqrt{2}$ is a very close approximation to $\frac{9\pi}{4}$. $\endgroup$
    – Jonathan Love
    Commented Oct 9, 2021 at 1:19

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