Before I ask my question, I'll lead up to it through an example. Just for reference, I have only taken up to Calc 3 and haven't taken Vector Calc. Lets look at this definite integral:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]∫∫cos(x^2+y^2)dxdy[/tex]

The bounds on the outer integral is from 0 to 1 while the bounds on the inner integral is from 0 to [itex]\sqrt{1-y^2}[/itex]. I don't know how to include that in the Latex. If you have taken Calc 3 or at least Calc 2, you will notice that this is an impossible integral to take in Cartesian coordinates. Anything learned in Calc 2(u sub, by parts, trig sub, etc...) to take this integral fails. However, if you switch to polar coordinates, it becomes possible to take this integral. You can see the full problem done out here in example 5. The polar form of integration can be derived from the Jacobian Matrix and it is simple to show this(in example 2).

Here is my question. There are a ton of integrals that look impossible to do in certain coordinate systems but if we switch coordinate systems, they become possible like in this example. Does this mean that all integrals are potentially possible to do if we switch coordinate systems by using Jacobian Matrices?

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# Integration and the Jacobian

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