anonymous one year ago How would I do this?

1. anonymous

2. ganeshie8

sub $$u=2x$$

3. anonymous

Hmm... but I don't have the function... Im kind of confused

4. ganeshie8

$\int\limits_0^2 f(2x)\,dx~~\stackrel{u=2x}{=}~~ \frac{1}{2}\int\limits_0^4 f(u)\,du = \frac{1}{2}*20=10$

5. anonymous

Why does the limit of integration change from 4 to 2?

6. ganeshie8

scratch that, lets work it again from beginning

7. ganeshie8

you want to evaluate $\int\limits_0^2f(2x)\, dx$ substitute $$u=2x \implies du=2dx\implies dx=\dfrac{du}{2}$$

8. ganeshie8

next work the bounds, as $$x\to 0$$, what does $$u\to$$ ? as $$x\to 2$$, what does $$u\to$$ ?

9. anonymous

1) x -> 0, u -> 0 2) x -> 2, u -> 4

10. ganeshie8

so upon substitution, bounds change from (0, 2) to (0, 4) and the differential changes from dx to du/2 plug them in

11. anonymous

okay, so then that's $\int\limits_{0}^{4}f(u) \frac{ du }{ 2 }$ and then the 1/2 comes out of the integral

12. ganeshie8

Yes, next recall that the variable in definite integral is "dummy" $\int\limits_a^b f(\color{red}{x})\,d\color{red}{x} = \int\limits_a^b f(\color{red}{t})\,d\color{red}{t}=\int\limits_a^b f(\color{red}{\spadesuit})\,d\color{red}{\spadesuit}$

13. anonymous

right

14. anonymous

Ohhh now I get what's happening!

15. anonymous

@ganeshie8 Thanks so much!