Final Answer:
The integral's value is -21, representing the signed area under a curve. This result implies that the net accumulation of the function's values over the specified interval is negative 21 units.
Step-by-step explanation:
Let's find the antiderivative of g(x) using the given values of g(1) and g(5). We can use the definition of an antiderivative as follows:
An antiderivative of a function f(x) is a differentiable function F(x) such that F'(x) = f(x).
Since we know the values of g(1) and g(5), we can find an antiderivative F(x) such that F'(x) = g(x). Let's call this function F(x).
Now, we can find F(x) by integrating g(x) using the given values. We have:
F(x) = -1 * x + C, where C is a constant of integration.
We know that F(1) = -1, which gives us:
-1 = -1 * 1 + C. Therefore, C = 0.
Similarly, we know that F(5) = -6, which gives us:
-6 = -1 * 5 + 0. Therefore, F(x) = -x - 6.
So, the value of the integral from 1 to 5 is:
∫^5_1 g(x) dx = F(5) - F(1) = (-6 - 1) - (-1 - 0) = -21.