Question

Please please help 100 points

Answer 1

We're given

$\displaystyle \int_4^{-10} g(x) \, dx = -3$

which immediately tells us that

$\displaystyle \int_{-10}^4 g(x) \, dx = 3$

In other words, swapping the limits of the integral negates its value.

Also,

$\displaystyle \int_4^6 g(x) \, dx = 5$

The integral we want to compute is

$\displaystyle \int_{-10}^6 g(x) \, dx$

which we can do by splitting up the integral at x = 4 and using the known values above. Then the integral we want is

$\displaystyle \int_{-10}^6 g(x) \, dx = \int_{-10}^4 g(x) \, dx + \int_4^6 g(x) \, dx = 3 + 5 = \boxed{8}$