1. ## Definite Integral

If the definite integral from -2 to 6 of f(x)dx=10 and the definite integral from 2 to 6 of f(x)dx=3, then the definite integral from 2 to 6 of f(x-4)dx= ?

I don't understand how to solve definite integrals when the function has something more than just x inside the parenthesis such as f(4-x).

2. Originally Posted by frog09
If the definite integral from -2 to 6 of f(x)dx=10 and the definite integral from 2 to 6 of f(x)dx=3, then the definite integral from 2 to 6 of f(x-4)dx= ?

I don't understand how to solve definite integrals when the function has something more than just x inside the parenthesis such as f(4-x).
Here is a hint: what does substituting $\displaystyle 4-x$ instead of $\displaystyle x$ do to the function? An easy example to find out is to graph $\displaystyle f(x) = x^2$ and then graph $\displaystyle f(x-4) = (x-4)^2$. Compare $\displaystyle \int_2^6 x^2 dx$ and $\displaystyle \int_6^{10} (x-4)^2 dx$

Do you see why? What I used a change of variables for the second integral, where $\displaystyle y=x-4,y+4=x$ and $\displaystyle dy = dx$?

Once you figure that out, recall that for that integral is a linear operator:
$\displaystyle \int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$ where $\displaystyle a<b<c \in \mathbb{R}$

You should get the answer to be 7: $\displaystyle \int_2^6 f(x-4) dx = 7$

3. Originally Posted by frog09
If the definite integral from -2 to 6 of f(x)dx=10 and the definite integral from 2 to 6 of f(x)dx=3, then the definite integral from 2 to 6 of f(x-4)dx= ?

I don't understand how to solve definite integrals when the function has something more than just x inside the parenthesis such as f(4-x).
Hint: Let $\displaystyle t = 4-x$ in $\displaystyle \int^6_2 f(4-x)\;dx.$ (Too slow)

4. Originally Posted by frog09
If the definite integral from -2 to 6 of f(x)dx=10 and the definite integral from 2 to 6 of f(x)dx=3, then the definite integral from 2 to 6 of f(x-4)dx= ?

I don't understand how to solve definite integrals when the function has something more than just x inside the parenthesis such as f(4-x).
$\displaystyle \int_{-2}^6 f(x) \, dx = 10$

$\displaystyle \int_2^6 f(x) \, dx = 3$

$\displaystyle \int_2^6 f(x-4) \, dx = \, ?$

use the method of substitution ...

$\displaystyle u = x - 4$

$\displaystyle du = dx$

substitute and reset the limits of integration ...

$\displaystyle \int_2^6 f(x-4) \, dx$

$\displaystyle \int_{-2}^2 f(u) \, du$

$\displaystyle \int_{-2}^6 f(x) \, dx - \int_2^6 f(x) \, dx = \int_{-2}^2 f(x) \, dx$

$\displaystyle 10 - 3 = 7$

since $\displaystyle \int_{-2}^2 f(u) \, du = \int_{-2}^2 f(x) \, dx$

$\displaystyle \int_{-2}^2 f(u) \, du = \int_2^6 f(x-4) \, dx = 7$

5. LOL @ danny arrigo

Sorry

6. I was re-checking this problem and realized I copied down the last part wrong. It is really supposed to be:
$\displaystyle \int_2^6 f(4-x) , dx = ?$

i substitute u=4-x and get -du=dx. and get the new limits as
$\displaystyle \int_2^{-2} f(u) du$

which would be
$\displaystyle -\int_{-2}^2 f(u) du$

does that -du change the problem? or would it still be 7?

7. Originally Posted by frog09
I was re-checking this problem and realized I copied down the last part wrong. It is really supposed to be:
$\displaystyle \int_2^6 f(4-x) , dx = ?$

i substitute u=4-x and get -du=dx. and get the new limits as
$\displaystyle \int_2^{-2} f(u) du$

which would be
$\displaystyle -\int_{-2}^2 f(u) du$

does that -du change the problem? or would it still be 7?
You would get a negative from the dx so your second last line should be - and the last line should be +.