Find the area enclosed between f(x)=05x2+3 and g(x)=x from x=−8 to x=3.

To find the area enclosed between two curves, we need to calculate the definite integral of the difference between the two functions over the given interval.

In this case, we need to find the area between f(x) = 0.5x^2 + 3 and g(x) = x from x = -8 to x = 3.

Step 1: Set up the integral
The area can be calculated using the integral:
A = ∫[a, b] |f(x) - g(x)| dx
In this case, a = -8 and b = 3, and the integrand is |f(x) - g(x)|.

Step 2: Calculate the difference between the two functions
f(x) - g(x) = (0.5x^2 + 3) - x
Simplifying, we get:
f(x) - g(x) = 0.5x^2 - x + 3

Step 3: Take the absolute value
Since the integrand involves the absolute value, we need to consider both positive and negative parts of the difference.

Case 1: f(x) - g(x) ≥ 0
In this case, we can leave the integrand as it is.

Case 2: f(x) - g(x) < 0
In this case, we need to take the negative of the integrand to ensure we get the correct area.

Step 4: Evaluate the integral
Now, we can evaluate the integral:
A = ∫[-8, 3] |f(x) - g(x)| dx
= ∫[-8, 3] (0.5x^2 - x + 3) dx

Integrating the function (0.5x^2 - x + 3) over the interval [-8, 3] will give us the area enclosed between the two curves.