Let be a positive real number. The parabolas defined by and intersect at the points and .
Prove that the area enclosed by the two curves is constant. Explain why.
First of all we note that
Let . Evaluate the sum
We have successively
Prove that the sum
is independent of .
Evaluate the integral
Let and . The Jacobian is
However , since we conclude that
where is Sophomore’s dream constant.
Evaluate the area of the given triangle: