## Constant area

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.

Solution

First of all we note that

Hence,

## Binomial sum

Let . Evaluate the sum

Solution

We have successively

## Independent of n

Prove that the sum

is independent of .

## Sophomore’s dream constant

Evaluate the integral

Solution

Let and . The Jacobian is

Hence,

However , since we conclude that

where is Sophomore’s dream constant.

## Area of triangle

Evaluate the area of the given triangle: