# Area Between Curves

1. Oct 12, 2011

### Rapier

1. The problem statement, all variables and given/known data

Find the area bounded by the curves y=x^2 and y= 2 - x^2 for 0 ≤ x ≤ 2.

2. Relevant equations

∫top - ∫bottom

3. The attempt at a solution

∫(2-x^2)dx - ∫x^2dx

What I'm confused about is that the two equations only cross on [-1,1] so within the interval of the problem I only have an enclosed area on [0,1]. But the problem asks for the area on [0,2]. How do I reconcile the differing intervals?

2. Oct 12, 2011

### LCKurtz

If you draw the vertical line x = 2 it gives a right boundary just like x = 0 gives the left boundary. Your curves cross so you have to do it in two parts.

3. Oct 12, 2011

### Rapier

OH! I think I see that now.

So I'll have:

[∫(0→1)(2-x^2)dx - ∫(0→1)x^2dx] + [∫(1→2)(2-x^2)dx - ∫(1→2)x^2dx]

Basically the area between the curves on [0,1] plus the bits hanging off on [1,2].

A = 4/3 un^2

I knew there was something I was missing and it's been a couple of weeks since we did that.

Thanks for the helps!

4. Oct 12, 2011

### LCKurtz

Your integrand is always y-upper - y-lower. Check that on the interval [1,2].

5. Oct 12, 2011

### Rapier

Oh! Yep. I forgot that my lines crossed.

One step at a time.... :)

A = 4 un^2

Thanks again.