Solving a Polar Coordinate Problem in Calculus.

In this calculus post, I am going to work through a fun problem that I received as an undergraduate. The question states:

Use polar coordinates to evaluate the double integral of the function

over the region determined by the equations

First of all, what I did was take the first equation and rewrite it in terms of x and y…

In terms of x:

In terms of y:

Then, I put the equations into Desmos and got the following graph:

Image 1

Using the equation given will make the integrals much harder to solve, therefore, I have shown in the image how we can change our given equation into simpler form (i.e polar coordinates). Instead of

we will use the polar coordinates:

Now that we have drawn our picture and understand the equations given, it is time to put it all together and begin solving the equation:

And we have our answer. The answer is negative because we were given that

And that concludes this cool problem. Thank you for reading.