How to use this function to evaluate this integral

[hrappur2]

Hey guys! I'm not from an English speaking country so I'll do my best to translate. I would appreciate if you could help me with this problem.

1. Homework Statement

Increasing (growing) function is f(0)=0 and f(a)=b and a,b>0
Use ∫(from 0 to a)f(x)dx=ab-∫(from 0 to b)f^-1(x)dx to solve ∫(from 0 to 2)arcsin(x/4)


3. The Attempt at a Solution

I know how to find ∫(from 0 to 2)arcsin(x/4) so that's no problem but I just don't understand how I use the other part to find it, what am I supposed to do? Help would be appreciated!

Thanks in advance!

 

[HallsofIvy]

Your formula says that
$$\int_0^a f(x)dx= ab- \int_0^b f^{-1}(x)dx$$

Here, $f(x)= arcsin(x/4)$ so what is $f^{-1}(x)$? a= 2 and b= arcsin(1/2). What is that?

 

[hrappur2]

Thanks for the reply! I understand now how to think this but I still get wrong answer. So a=2 and therefore b must be arcsin(2/4) like you said. f(x)^-1 is sin(x/4).

When I put all this into the function I get 2arcsin(1/2) - ∫(from 0 to arcsin(1/2))sin(x/4) = 1.01...

But when I calculate ∫(from 0 to 2)arcsin(x/4) I get 0.51...

What am I doing wrong?

EDIT: OK never mind, I got the right answer. Thanks for your help!