Integrating with Changing Intervals: Finding the Area Between Two Curves

[titasB]

Homework Statement

Find ∫ f(x) dx between [4,8]

if,

∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4

Homework Equations

[/B]
∫ f(x) dx between [4,8] ,
∫ f(2x) dx between [1,4] = 3 and ∫ f(x) dx between [2,4] = 4

The Attempt at a Solution

We are given ∫ f(2x) dx between [1,4] = 3

Let, 2x = u ⇒ dx = du/2

So, the new intervals are u2 = 2(4) = 8 and u1 = 2(1) = 2

This gives: 1/2 ∫ f(u) du between [2,8] = 3 ⇒ ∫ f(u) du between [2,8] = 6

And so to find t ∫ f(x) dx between [4,8]

I subtract ∫ f(x) dx between [2,4] from ∫ f(2x) dx between [1,4]

which is the same as writing: ∫ f(u) du between [2,8] - ∫ f(x) dx between [2,4] = 6- 4 = 2

Is this the correct answer? I'm not sure if ∫ f(u) du between [2,8] = 6 is the same as ∫ f(x) dx between [2,8] = 6
I read something about a dummy variable and this seems like a reasonable answer. Please let me know.

 

[wabbit]

Yes your reasoning and result are correct. And indeed x or u are just placeholders in the integrals, you can replace them with any symbol you like.

 

[titasB]

thank you