Definite integral

[sylvos]

Can anyone help me with this puzzling issue?
SO the definite integral from -2 to 4 of f(x)dx = 5, and from -2 to 0 = 5 and from 2 to 4 = 2
I need to find the definite integral from 0 to 2 of f(x)dx = ?
and from 2 to 0 of (5f(x)-5)dx = ?

I've already figured that the first one is equal to -2 since I can view 0 and 4 as the same and then its just a flipped 2 to 4 so it becomes the negative, but I can't figure out this last one. Any idea?

 

[sylvos]

Ok I solved the second problem, once again I assumed that f(0)=f(4) and since that's given as 2, I multiplied by 5 and subtracted the definite integral from 2 to 0 of 5 = -10 so I get 10-(-10)=20

 

[Deleted member 4993]

Can anyone help me with this puzzling issue?
SO the definite integral from -2 to 4 of f(x)dx = 5, and from -2 to 0 = 5 and from 2 to 4 = 2
I need to find the definite integral from 0 to 2 of f(x)dx = ?
and from 2 to 0 of (5f(x)-5)dx = ?

assuming the function is continuous within the domain[-2,4]

\(\displaystyle \int_{-2}^4f(x) dx \ - \ \int_{-2}^0f(x) dx \ - \ \int_{2}^4f(x) dx \ = \ \int_{0}^2f(x) dx \ = \ 5 - 5 - 2 \ = \ -2\)

\(\displaystyle \ \int_{2}^0f(x) dx \ = \ - \int_{0}^2f(x) dx \ = \ 2\)

\(\displaystyle \ \int_{2}^0[5f(x)-5] dx \ = \ \ 5\int_{2}^0f(x) dx \ - \ 5\ \int_{2}^0 dx \ = \ 10 - 5(0-2) \ = \ 20\)

I've already figured that the first one is equal to -2 since I can view 0 and 4 as the same and then its just a flipped 2 to 4 so it becomes the negative, but I can't figure out this last one. Any idea?