use the midpoint rule with the given value of n to approximate the integral. round the answer to four decimal places.

integral 0 to pi/2 2cos^3(x)dx, n=4

M4=????

To use the midpoint rule, we need to divide the given interval [0, π/2] into equally spaced subintervals. In this case, since n = 4, we will have 4 subintervals.

Step 1: Calculate the width of each subinterval (Δx).
Δx = (b - a)/n
Δx = (π/2 - 0)/4
Δx = π/8

Step 2: Find the midpoints of each subinterval.
x1 = 0 + Δx/2 = π/16
x2 = π/8 + Δx/2 = 3π/16
x3 = π/4 + Δx/2 = 5π/16
x4 = 3π/8 + Δx/2 = 7π/16

Step 3: Evaluate the function at each midpoint and calculate the sum.
M4 = Δx[f(x1) + f(x2) + f(x3) + f(x4)]
M4 = (π/8)[2cos^3(π/16) + 2cos^3(3π/16) + 2cos^3(5π/16) + 2cos^3(7π/16)]

Now, we substitute the values into the equation and round the answer to four decimal places.

M4 ≈ (π/8)[2cos^3(π/16) + 2cos^3(3π/16) + 2cos^3(5π/16) + 2cos^3(7π/16)]
≈ (π/8)[2cos^3(π/16) + 2cos^3(3π/16) + 2cos^3(5π/16) + 2cos^3(7π/16)]
≈ (π/8)[2(0.9761) + 2(0.6801) + 2(0.2821) + 2(0.5777)]
≈ (π/8)[1.9522 + 1.3602 + 0.5642 + 1.1554]
≈ (π/8)(4.0319)
≈ (3.1416/8)(4.0319)
≈ 1.5708(4.0319)
≈ 6.3138

Therefore, M4 ≈ 6.3138 rounded to four decimal places.

To approximate the integral using the midpoint rule, we first need to divide the interval [0, π/2] into smaller subintervals. The value of n determines the number of subintervals. In this case, n = 4 implies we need to divide [0, π/2] into four equal subintervals.

Step 1: Calculate the width of each subinterval.
The width of each subinterval is determined by dividing the total width of the interval by the number of subintervals. In this case, the width of each subinterval is (π/2 - 0)/4 = π/8.

Step 2: Determine the midpoint of each subinterval.
To use the midpoint rule, we need to find the x-coordinate of the midpoint of each subinterval. Starting from 0, we add half of the width to each subsequent midpoint until we reach π/2. Since we have four subintervals, we need to find the values of x at x = π/8, 3π/8, 5π/8, and 7π/8.

Midpoint 1: x = (0 + π/8)/2 = π/16
Midpoint 2: x = (π/8 + 3π/8)/2 = π/4
Midpoint 3: x = (3π/8 + 5π/8)/2 = π/2
Midpoint 4: x = (5π/8 + 7π/8)/2 = 3π/4

Step 3: Evaluate the function at each midpoint.
Now that we have the x-coordinates of each midpoint, we substitute these values into the function to find the corresponding function values.

Function value at x = π/16: 2cos^3(π/16)
Function value at x = π/4: 2cos^3(π/4)
Function value at x = π/2: 2cos^3(π/2)
Function value at x = 3π/4: 2cos^3(3π/4)

Step 4: Calculate the sum of the function values times the width of each subinterval.
To approximate the integral using the midpoint rule, we sum up the product of each function value and the width of the subinterval to obtain the approximation.

Approximation = (π/8) * [2cos^3(π/16) + 2cos^3(π/4) + 2cos^3(π/2) + 2cos^3(3π/4)]

Using a calculator to evaluate the sine and cosine values, we can calculate the approximation by substituting the values and rounding to four decimal places.

delta x = pi/8

x at pi/16 , 3pi/16, 5pi/16, 7pi/16
so
2 cos^3 pi/16 = 1.887
2 cos^3 3 pi/16 = 1.1497
etc
add them up and multiply by pi/8