Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.
int_2^10 2 sqrt(x^2+5)dx text(, ) n=4
Explain what you mean by "n" and "text(, )"
Probably no one has responded to your questions because your notation is unfamiliar.
To approximate the integral using the Midpoint Rule with n = 4, follow these steps:
Step 1: Calculate the width of each subinterval.
The width, Δx, can be found by dividing the length of the interval by the number of subintervals:
Δx = (b - a) / n
In this case, a = 2 and b = 10.
Δx = (10 - 2) / 4
Δx = 8 / 4
Δx = 2
Step 2: Find the midpoint of each subinterval.
To find the midpoint of each subinterval, add Δx/2 to the starting point of each subinterval. We start with x = 2 and the subintervals are of width 2, so the midpoints are:
x1 = 2 + (2/2) = 3
x2 = 3 + (2/2) = 4
x3 = 4 + (2/2) = 5
x4 = 5 + (2/2) = 6
Step 3: Evaluate the function at each midpoint.
For each midpoint, evaluate the function 2√(x^2 + 5).
At x = 3:
f(3) = 2√(3^2 + 5) = 2√(9 + 5) = 2√(14)
At x = 4:
f(4) = 2√(4^2 + 5) = 2√(16 + 5) = 2√(21)
At x = 5:
f(5) = 2√(5^2 + 5) = 2√(25 + 5) = 2√(30)
At x = 6:
f(6) = 2√(6^2 + 5) = 2√(36 + 5) = 2√(41)
Step 4: Calculate the sum of the function values multiplied by Δx.
Add up the function values evaluated at the midpoints and multiply each by Δx, then sum them up:
Approximation = Δx * (f(3) + f(4) + f(5) + f(6))
= 2 * (2√(14) + 2√(21) + 2√(30) + 2√(41))
You can now compute the approximation using a calculator or software and round the answer to four decimal places.
To approximate the integral using the Midpoint Rule, we need to divide the interval [2, 10] into n equal subintervals. In this case, n is given as 4.
Step 1: Find the width of each subinterval.
Width = (b - a) / n
Width = (10 - 2) / 4
Width = 8 / 4
Width = 2
Step 2: Find the midpoints of each subinterval.
For n = 4, we have 4 subintervals. The midpoints are calculated by adding half of the width to the left endpoint of each subinterval.
- Subinterval 1: x = 2 + (1/2) * 2 = 2 + 1 = 3
- Subinterval 2: x = 2 + (1/2) * 2 * 2 = 2 + 2 = 4
- Subinterval 3: x = 2 + (1/2) * 2 * 3 = 2 + 3 = 5
- Subinterval 4: x = 2 + (1/2) * 2 * 4 = 2 + 4 = 6
Step 3: Evaluate the function at the midpoints.
Evaluate the function, 2√(x^2 + 5), at each midpoint.
- f(3) = 2√(3^2 + 5) = 2√(9 + 5) = 2√14
- f(4) = 2√(4^2 + 5) = 2√(16 + 5) = 2√21
- f(5) = 2√(5^2 + 5) = 2√(25 + 5) = 2√30
- f(6) = 2√(6^2 + 5) = 2√(36 + 5) = 2√41
Step 4: Calculate the sum of the function values multiplied by the width.
Approximation = Width * [f(midpoint1) + f(midpoint2) + f(midpoint3) + f(midpoint4)]
Approximation = 2 * [2√14 + 2√21 + 2√30 + 2√41]
Now, you can compute the numerical value of the integral by substituting the function values and calculating the sum. Finally, round the answer to four decimal places.
In general midpoint rule is
integral from
Xo to Xn is approximated by rectangles of base delta x = Xk - Xk-1 and height f([Xk+Xk-1]/2)
so
integral is approximately
sum from k = 1 to k = n of f([Xk+Xk-1]/2) delta x
NOW HERE - You have not told us over what interval
nor do I really understand what f(x) is
like what does 2^10 2 mean?
2^10 is 2 to the tenth. Is this times another two so it is 2^11. Please be careful with parentheses.
Anyway here is how to do the problem.
Divide whatever your interval is by four if n is four then delta x = interval/4
label those five x points Xo X1 X2 X3 X4
calculate f(x) at (X1-Xo)/2 and (X2-X1)/2 and (X3-X2)/2 and (X4-X3)/2
then add them up and multiply by delta x
done