Final answer:
To approximate the integral of ln(8 + e^x) from 0 to 4, we use numerical methods: the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule, all with n=8, applying different formulas to function evaluations at specific interval points.
Step-by-step explanation:
The question asks to approximate the integral of ln(8 + ex) from 0 to 4 using the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule with n = 8. The first step is to calculate the width of each interval, which is (b-a)/n, where a = 0, b = 4, and n = 8. For all three rules, we need to evaluate the function at specific points and then apply the respective formulas for each numerical integration method.
Trapezoidal Rule
Add the value of the function at the beginning and end, plus twice the value of the function at points within the interval (excluding the end points), multiply by the width of the interval divided by 2.
Midpoint Rule
Calculate the midpoint of each subinterval, evaluate the function at these midpoints, sum these evaluations, and multiply by the width of the interval.
Simpson's Rule
Add the function's value at the endpoints, four times the function value at the odd intervals, and two times the function value at the even intervals except for the last point. Then, multiply by the width of the interval divided by 3.