Refer to the attachment: For the part that is underlined in red, on the right hand side of the equation why does the integral's limits go from -5 to 100 + 100 to 12? It isnt going in sequence (due to the 100 to 12) and that doesnt make sense
Refer to the attachment: For the part that is underlined in red, on the right hand side of the equation why does the integral's limits go from -5 to 100 + 100 to 12? It isnt going in sequence (due to the 100 to 12) and that doesnt make sense
Because it is given in the question.
We are told
$\displaystyle \int_{100}^{-5} f(x) dx = 4 $
Which is the same thing as saying
$\displaystyle - \int_{-5}^{100} f(x) dx = 4 $
Going back to the question we want
$\displaystyle \int_{-5}^{12} f(x) dx $
Well...we know the value from -5 to 100, so lets use that. But to make this equal, we then need to go from 100 to 12! Therefore, we get
$\displaystyle \int_{-5}^{12} f(x) dx = \int_{-5}^{100}f(x)dx + \int_{100}^{12} f(x) dx $
We already know $\displaystyle \int_{-5}^{100} f(x)dx = -4 $ but we don't quite know the integral of 100 to 12...so let's break that guy up in the same way.
$\displaystyle \int_{100}^{12} f(x) dx = \int_{100}^{-10} f(x)dx + \int_{100}^{12} f(x) dx $
Of course by reversing the limits we can find a numerical value.
To re-state...we choose our limits given the definitions in the original question