# Integration Difficulty It is very easy to calculate the area of a circle via polar coordinates (or breaking the circle up into cones and summing them up all). .png) This then lead me to ask the question: Is there a straightforward way to calculate the area of the circle via cartesian coordinates? Well, after trying many times and breaking it down into a Riemann sum and taking the limit as n -> infinity, it doesn't look like there is an _easy_ way to do this. I just end up with a nasty limit: .png) Well, this got me thinking, the shape and area of of $1 - x^2$ is somewhat similar to the top half of a circle:  And I can _easily_ find the area under this curve from -1 to 1, via either a standard integration or computing the actual underlying Riemann summation (I arrive at the correct value of $\frac{4}{3}$ in both cases: .png) #### Question 1 These shapes are "similar" from a high level. They both have clear areas under the curve. And, when I break them up into finer and finer partitions, I can compute the integral numerically (in a cartesian coordinate system) with ease. Yet, the circle proves to be _tremendously difficult_ to calculate via traditional integration in a cartesian coordinate system. What is it about using polar coordinates that makes this so much easier? #### Answer 1 My intuition is that using polar coordinates we are implicitly encoding a great deal of information about _pi_ and specifically the _circle_ itself. This information allows us to much more easily represent area elements of the circle (in my case I used cone area elements), which greatly simplifies our problem. So by changing the way we _describe_ our function (top half of a circle) in a way that has information about _circles_ (which polar coordinates clearly do), we have in a sense provided context/information that the integration process can take advantage of? Answer from Matt Grimes: Yep! It's encoded in what's called the "volume form." In cartesian coordinates, you get area by integrating 1dxdy, but in polar coordinates you get area by integrating $rdr d \theta$--that's encoded the area of a circle, so we can't use polar coordinates to **derive** the area of a circle. Note that $r$ here is the determinant of the Jacobian of the coordinate change to polar: $x = rcos(\theta)$, $y = r sin(\theta)$. #### Question 2 Why is it that some limits are incredibly hard to solve for mathematically, but very straight forward to implement and show convergence computationally?  #### Answer 2 Matt Grimes response: > For the limits thing...I guess I'd say that your phrase "show convergence computationally" is doing a lot of lifting. You need a few theoretical tools to argue that the approximations have significance, that they converge, etc. So for instance, unless you can argue that the sequence is "Cauchy" then it's not clear that going further in the loop won't give wildly different answers. maybe to answer your question about how I think about it: I see a very strong distinction between: > * convince yourself something is true > * establish that something is true