Question

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. INT from 0 to pi/2 (2cos5 x) dx, n = 4

Answer 1

To approximate the integral using the Midpoint Rule with n = 4, divide the interval [0, pi/2] into 4 subintervals. Evaluate the function at the midpoint of each subinterval and calculate the sum of the areas of rectangles. The approximate value of the integral is -0.1941.

Step-by-step explanation:

To approximate the integral using the Midpoint Rule, we need to divide the interval [0, pi/2] into equal subintervals. Since n = 4, we have 4 subintervals: [0, pi/8], [pi/8, pi/4], [pi/4, 3pi/8], and [3pi/8, pi/2]. The Midpoint Rule states that the integral is approximated by the sum of the areas of rectangles with height equal to the function evaluated at the midpoint of each subinterval.

Let's calculate the approximate value of the integral:

1. Subinterval 1: [0, pi/8]. The midpoint is (0 + pi/8)/2 = pi/16. Evaluate the function at this point: 2cos(5(pi/16)) = 1.9696.
2. Subinterval 2: [pi/8, pi/4]. The midpoint is (pi/8 + pi/4)/2 = 3pi/16. Evaluate the function at this point: 2cos(5(3pi/16)) = 0.3124.
3. Subinterval 3: [pi/4, 3pi/8]. The midpoint is (pi/4 + 3pi/8)/2 = 5pi/16. Evaluate the function at this point: 2cos(5(5pi/16)) = -0.7391.
4. Subinterval 4: [3pi/8, pi/2]. The midpoint is (3pi/8 + pi/2)/2 = 13pi/32. Evaluate the function at this point: 2cos(5(13pi/32)) = -1.8773.

The approximate value of the integral is the sum of the areas of these rectangles: 1.9696(delta_x) + 0.3124(delta_x) - 0.7391(delta_x) - 1.8773(delta_x), where delta_x is the width of each subinterval. Since n = 4, delta_x = (pi/2 - 0)/4 = pi/8.

Simplifying the expression, we have: 1.9696(pi/8) + 0.3124(pi/8) - 0.7391(pi/8) - 1.8773(pi/8) = -0.6182(pi/8) = -0.1941.

Therefore, the approximate value of the integral using the Midpoint Rule with n = 4 is -0.1941.