Verified answer

I will assume you mean the integral from -4 to 4 of the given function -- it is a little hard to tell from your notation. To make it a little easier, note that your function is even so we can just double the integral from 0 to 4.

f(x) = x^4 - 17 x^2 + 16 so the fundamental theorem of calculus states that we need the antiderivative of this function which is:

x^5/5 - 17 x^3/3 + 16 x and subract the value of this function at the x=0 from the value at x=4. But the value at x=0 is zero so our final answer is simply:

2 ( 4^5/5 - 17*4^3/3 + 16*4) =

2 ( 1024/5 - 17*64/3 + 64) =

2 ( (3072 - 5440 + 960)/15) =

2( -1408/15) = - 2816/15

This represents the "net" area under the curve between the given limits because the function is both positive and negative so it cancels. To get the actual area you would have to integrate the absolute value of the function.