How Do You Solve the Areas Between Curves Problem Involving Unknown Functions?

[robertmatthew]

Homework Statement

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 [y^1/2-(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

Homework Equations

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

The Attempt at a Solution

The first thing I did was try to find the area A.
A= int 0-->a² [y^1/2-(y/2)^1/2] dy
A= (2/3)y^3/2 - (1/sqrt2)(2/3)y^3/2| 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?

 

[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.

 

[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)?

 

[Simon Bridge]

Are the other conditions f(0)=0 and f(x)≤x²?

Yes.

I don't understand how to verify those, would it have to do with the graph of the region?

Yes.

Since A=B and for A, both of the curves have a point at (0,0) so B must have two points there also?

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

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

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.