Final answer:
To estimate the integral using the Trapezoidal Rule and the Midpoint Rule, both with n=4, we divide the interval into four equal parts, calculate the function values at endpoints and midpoints, and apply the respective formulas.
Step-by-step explanation:
To estimate the integral ℓ01 cos(x2) dx using the Trapezoidal Rule and the Midpoint Rule with n = 4, we first need to partition the interval [0, 1] into four subintervals of equal length, which means each subinterval has a length of Δx = 1/4. Then we will calculate the approximation using the formulas for both rules.
Trapezoidal Rule
For the Trapezoidal Rule, the approximation is given by:
Approximation = (Δx/2) × [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)]
Midpoint Rule
For the Midpoint Rule, the approximation is calculated by:
Approximation = Δx × [f((x0+x1)/2) + f((x1+x2)/2) + f((x2+x3)/2) + f((x3+x4)/2)]
To perform the actual computation, we plug in the values of x that correspond to the endpoints and midpoints of the subintervals into the function cos(x2) and then apply the respective formulas.