Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
interval [0,72] sin square root of xdx
where n=4
To approximate the integral using the Midpoint Rule with n=4, we need to divide the given interval [0, 72] into 4 subintervals of equal width.
Step 1: Calculate the width, Δx, of each subinterval:
Δx = (b - a) / n
= (72 - 0) / 4
= 18
Step 2: Calculate the midpoint, xi, of each subinterval:
x1 = 0 + (Δx / 2) = 0 + (18 / 2) = 9
x2 = 0 + (Δx / 2) + Δx = 9 + 18 = 27
x3 = 0 + (Δx / 2) + 2Δx = 9 + 36 = 45
x4 = 0 + (Δx / 2) + 3Δx = 9 + 54 = 63
Step 3: Evaluate the integrand, sin(sqrt(x)), at each midpoint:
f(x1) = sin(sqrt(9))
f(x2) = sin(sqrt(27))
f(x3) = sin(sqrt(45))
f(x4) = sin(sqrt(63))
Step 4: Calculate the sum of the products of the function values and the width of each subinterval:
Approximation = Δx * (f(x1) + f(x2) + f(x3) + f(x4))
= 18 * (sin(sqrt(9)) + sin(sqrt(27)) + sin(sqrt(45)) + sin(sqrt(63)))
Finally, plug in the values of sin(sqrt(9)), sin(sqrt(27)), sin(sqrt(45)), and sin(sqrt(63)) into a calculator, and perform the calculations to obtain the approximated value of the integral rounded to four decimal places.
To approximate the integral using the Midpoint Rule, we divide the interval into equal subintervals, each of width Δx. The formula for the Midpoint Rule is as follows:
∫f(x)dx ≈ Δx * (f(x₁/2) + f(x₃/2) + f(x₅/2) + ... + f(xₙ₋₁/2)),
where f(x) is the function we are integrating, Δx is the width of each subinterval (which can be calculated by Δx = (b-a)/n, where a and b are the limits of the interval, and n is the given value), and x₁/2, x₃/2, x₅/2, ..., xₙ₋₁/2 are the midpoints of each subinterval.
In this case, the given interval is [0, 72], and n = 4. We need to find the value of Δx and evaluate the function at the midpoints.
Δx = (b-a)/n = (72-0)/4 = 18.
To find the midpoints, we need to identify the x-coordinates at the centers of each subinterval. Since we have 4 subintervals, the midpoints would be as follows:
x₁/2 = 0 + Δx/2 = 0 + 18/2 = 9,
x₃/2 = 9 + Δx = 9 + 18 = 27,
x₅/2 = 27 + Δx = 27 + 18 = 45,
x₇/2 = 45 + Δx = 45 + 18 = 63.
Now that we have the midpoints, we can evaluate the function f(x) = sin(sqrt(x)) at each of these points:
f(9) ≈ sin(sqrt(9)),
f(27) ≈ sin(sqrt(27)),
f(45) ≈ sin(sqrt(45)),
f(63) ≈ sin(sqrt(63)).
Finally, substitute these values into the Midpoint Rule formula:
∫[0,72] sin(sqrt(x))dx ≈ 18 * [f(9) + f(27) + f(45) + f(63)].
Calculate the approximated value of the integral by evaluating the function at these midpoints, add up the results, and then multiply the sum by 18.
Make sure to round the answer to four decimal places.