# Areas between Curves problem

1. Feb 27, 2014

### robertmatthew

1. The problem statement, all variables and given/known data
Let a > 0 be a fixed real number. Define A to be the area bounded between y=x2,y=2x2, and y=a2. Define B to be the area between y = x2, y = f(x), and x = a where f(x) is an unknown function.
a) Show that if f(0) = 0, f(x) ≤ x2, and A = B then

int 0-->a2 [y1/2-(y/2)1/2] dy = int 0--> a [x2 - f(x)] dx

b) Find f(x) under the conditions above are true for all a > 0

2. Relevant equations

I guess the equations for the lines, y=x2, y=2x2, y=a2
and y=x2, y=f(x), and x=a

3. The attempt at a solution

The first thing I did was try to find the area A.
A= int 0-->a2 [y1/2-(y/2)1/2] dy
A= (2/3)y3/2 - (1/sqrt2)(2/3)y3/2| eval. 0 and a2
A= (2/3)a3 - (1/sqrt2)(2/3)a3
A= a3((2/3) - (1/sqrt2)(2/3))
a= (2/3)a3(1- (1/sqrt2))

Then I tried to find the area B, but this is where I got confused.
B= int 0--> a [x2 - f(x)] dx
B= [(1/3)x3|eval. 0-->a] - [int 0-->a [f(x)]]

I don't see how I can go any farther, because I don't think I can integrate an unknown f(x). Am I going about this the wrong way?

2. Feb 27, 2014

### Simon Bridge

The relation does not require you to interpret the unknown f(x).
Take another look at what you have to show.

If A=B then the relation drops out. The trick is to verify the other two conditions.

3. Feb 27, 2014

### robertmatthew

Are the other conditions f(0)=0 and f(x)≤x2? I don't understand how to verify those, would it have to do with the graph of the region? Since A=B and for A, both of the curves have a point at (0,0) so B must have two points there also?

And how should I go about part b, do I just solve for f(x)?

4. Feb 27, 2014

### Simon Bridge

Yes.

Yes.

No.
A=B just means the areas are the same - the graphs do not have to have points in common for that to happen.

i.e. what would happen to the RHS if f(x) > x^2

You should make sure you understand part (a) first, but basically you can separate out the integral in f(x) and use your understanding of the properties of integrals.