Final answer:
To approximate the definite integral using the Trapezoid Rule with n = 4, divide the interval into subintervals and apply the formula. Finally, we can evaluate the expression: Approximation = -204
Step-by-step explanation:
To approximate the definite integral ∫(x³ - 5x² + 5x + 5) dx from 0 to 8 using the Trapezoid Rule with n = 4, we need to divide the interval [0, 8] into 4 equal subintervals. The width of each subinterval is given by h = (8 - 0) / 4 = 2. The formula for approximating the integral using the Trapezoid Rule is:
Approximation = h/2 * [f(0) + 2f(2) + 2f(4) + 2f(6) + f(8)]
First, let's evaluate the function f(x) = x³ - 5x² + 5x + 5 at the given points:
f(0) = 5
f(2) = 3
f(4) = -11
f(6) = -29
f(8) = -45
Now we can substitute these values into the formula:
Approximation = 2/2 * [5 + 2(3) + 2(-11) + 2(-29) + (-45)]
Calculating the expression inside the brackets and simplifying gives:
Approximation = 2 * [-102]
Finally, we can evaluate the expression:
Approximation = -204