Undergrad How to calculate the area under a curve

[JG FRANKO]

I was doing some integral exercises for getting area under the function. I was doing only more simple stuff, like functions that don't go over the same "x area" multiple times, like a quadratic function. My question is how to calculate area of a loop in an equation x^3+y^3-3axy=0 if let's say a is 1.

Thank you for your answers in advance

 

[JG FRANKO]

If I create a function out of this equation, I get

Would the answer be correct, if I would use a definite integral to integrate this function to get the area of the loop?

 

[LCKurtz]

That curve is known as the Folium of Descartes. You can read about it, including how to find the area of that loop here:
https://en.wikipedia.org/wiki/Folium_of_Descartes

 

[Buffu]

If I create a function out of this equation, I get
View attachment 207587
Would the answer be correct, if I would use a definite integral to integrate this function to get the area of the loop?
View attachment 207588

The function you have will certainly won't give you the answer if you integrate it alone. You need a function for top part of the loop and one for the bottom part of the loop. On the top part the function is increasing and decreasing on the bottom part.