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# Tag Archives: Real Analysis

## 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

## 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.

## Differential equation (II)

Let be a differentiable function such that and

Find an explicit formula for .

**Solution**

Let us consider the function which is clearly differentiable. Hence,

Thus,

## Differential equation (I)

Let be a differentiable function such that and

Find an explicit formula for .

**Solution**

We have successively

which satisfies the given conditions.