I'm not sure how to do this definite integral

Question

If f(x) is even and ∫-6 to 6 (f(x) − 9) dx = 8, find ∫0 to 6 of f(x) dx.

I said 4 and then 13 but those were both wrong. I'm not sure how to account for the -9

Answer 1

Since f(x) is an even function, the graph is symmetric with respect to the y-axis.  

 

So, ∫_{(from -6 to 6)}f(x)dx = 2∫_{(from 0 to 6)}f(x)dx.

 

Therefore, ∫_{(from -6 to 6)}(f(x) - 9)dx = ∫_{(from -6 to 6)}f(x)dx - ∫_{(from -6 to 6)}9dx

 

                                                         2∫_{(from 0 to 6)}f(x)dx - 108 = 8

 

                                                         2∫_{(from 0 to 6)}f(x)dx = 116

 

                                                     So, ∫_{(from 0 to 6)}f(x)dx = 58