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So, if the Archimedian spiral is given with formula $r=2\theta$, what does that formula represent and what is the area of one turning of the spiral? The teacher solved it like: $$P=\frac{1}{2}\int_0^{2\pi}a^2\theta^2d\theta=\frac{1}{2}a^2\frac{\theta^3}{3}|_0^{2\pi}=\frac{8a^2\pi^3}{6}=\frac{4}{3}a^2\pi^3$$Can you please explain why he integrates $\frac{1}{2}a^2\theta^2$?

A6SE
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    Look at equation (8) here: http://mathworld.wolfram.com/PolarCoordinates.html – Jack D'Aurizio Mar 29 '15 at 14:29
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    Your teacher is finding the area for $r=a\theta$ – Ross Millikan Mar 29 '15 at 14:35
  • So, the equation (8) actually shows the area of a circle if r=r (radius) of a circle and $\theta1=0$ and $\theta2=2\pi$. The part I don't understand now is the equation of the spiral. What does $a^2$ represent? – A6SE Mar 29 '15 at 14:37
  • @A6Tech: Equation (8) would give the area of a disk if $r$ were a constant function of $\theta$ (namely, $r = R$ for a disk of radius $R$). In your situation, $r = a\theta$; the constant $a$ is a quantitative measure of "how rapidly" the spiral opens as the polar angle $\theta$ increases. – Andrew D. Hwang Mar 29 '15 at 15:28

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An intuitive derivation of the area element in polar coordinates goes as follows: imagine an infinitesimal pie slice with radius $r$ and angle $d \theta$. It is essentially an isosceles triangle with altitude $r$ and base $r\ d\theta$, so the area is $\frac 12r^2 \ d\theta,$ which is what your teacher is integrating.

Ross Millikan
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  • I understand for a circle, it makes sense, but what does $a$ represent? How does it change the circle to spiral? – A6SE Mar 29 '15 at 14:42
  • It is like integrating in Cartesian coordinates where the area element is $y\ dx$. You insert the expression for $y$ in terms of $x$ and integrate. Here your expression is $r=a\theta$ so we substitute that in, getting $\frac 12 (a\theta)^2\ d\theta$ as what we want to integrate. – Ross Millikan Mar 29 '15 at 14:45