Final answer:
The integral ∫[4,-4] f(x) dx results in a total area of 58⅓ by separately integrating the constant and quadratic parts of the piecewise function and summing them.
Step-by-step explanation:
Evaluation of the Integral
To evaluate the integral ∫[4,-4] f(x) dx where the function f(x) is piecewise defined as f(x) = 4 for -4 ≤ x ≤ 0 and f(x) = 16 - x² for 0 < x ≤ 4, we need to split the integral into two parts and calculate each separately.
The integral from -4 to 0 is straightforward since the function is a constant:
∫[0,-4] 4 dx = 4x | from -4 to 0 = 4(0) - 4(-4) = 16.
For the integral from 0 to 4, we need to integrate the quadratic term:
∫[4,0] (16 - x²) dx = 16x - x³/3 | from 0 to 4 = (16(4) - 4³/3) - (16(0) - 0³/3) = 64 - 64/3.
Adding both parts gives us the total area under the curve from x = -4 to x = 4:
16 + (64 - 64/3) = 16 + (192/3 - 64/3) = 16 + 128/3 = 16 + 42⅓ = 58⅓.
The integral is therefore 58⅓.