We are better than machines!

In Worksheet of problem 6 (problem 6) (we did not get as far), there is a challenging problem. It is interesting that a machine also has problems. Mathematica can not solve the problem!

Integrate[ 2x Cos[y-y^3/3],{x,0,1},{y,-Sqrt[1-x^2],0}] 

Of course, it can do so numerically

NIntegrate[ 2x Cos[y-y^3/3],{x,0,1},{y,-Sqrt[1-x^2],0}]

This gives a value 0.61837.... But we humans are more powerful and can get to the bottom of what that numbers are. Here are the two pages from my slides (not shown in class). If we change the order of integration and slice in an other direction, we can solve the integral and get sin(2/3). Indeed this is 0.61837...