Given a function f(x) = squr(3 + 2x), on [-1,1], subdivided into 4 equally large intervals, calculate the Riemann sum SR4 = 4Ei=1 f(xi)delta(x).
my attempt at the solution:
[-1,1], a = -1 and b = 1
delta(x) = 1 - (-1)/n = 2/n
xi = a + delta(x)i
xi = -1 + 2/n(i)
so after adding them up I get
40/n^2 - 8/n
Could someone please verify if this is correct?
Thank you
To calculate the Riemann sum SR4 for the function f(x) = sqrt(3 + 2x) on the interval [-1, 1], subdivided into 4 equally large intervals, you need to follow these steps:
Step 1: Determine the interval length (delta(x)):
The interval [-1, 1] is subdivided into 4 intervals, so the interval length can be calculated as:
delta(x) = (b - a) / n
where a = -1 is the lower limit, b = 1 is the upper limit, and n = 4 is the number of intervals.
delta(x) = (1 - (-1)) / 4 = 2 / 4 = 0.5
Step 2: Find the values xi for each interval:
xi can be found using the formula:
xi = a + delta(x) * i
where i is the index of the interval, ranging from 1 to n.
For n = 4:
xi = -1 + 0.5 * i
xi = -1 + 0.5i
Step 3: Calculate f(xi) for each xi:
Evaluate the function f(x) = sqrt(3 + 2x) for each xi calculated in Step 2.
For example, f(x1) = sqrt(3 + 2 * (-1 + 0.5 * 1)) = sqrt(3 + 2 * (-0.5)) = sqrt(3 - 1) = sqrt(2)
Similarly, calculate f(xi) for i = 2, 3, and 4.
Step 4: Calculate the Riemann sum SR4:
The Riemann sum SR4 is given by the formula:
SR4 = Σ f(xi) * delta(x)
where Σ represents the summation symbol.
Substituting the values of f(xi) and delta(x) obtained from Steps 3 and 1:
SR4 = 4 ∑(i=1 to 4) f(xi) * delta(x)
SR4 = 4 * (f(x1) * delta(x) + f(x2) * delta(x) + f(x3) * delta(x) + f(x4) * delta(x))
Substituting the values of f(xi) and delta(x):
SR4 = 4 * (sqrt(2) * 0.5 + ... + f(x4) * 0.5)
Now you need to substitute the values of f(xi) and calculate the sum.
Your attempt:
You wrote the formula correctly and calculated the values of xi correctly, but you made a mistake in the calculation of the sum. Instead of providing the sum equation, you only calculated the sum of certain terms.
To find the correct value for the Riemann sum SR4, substitute the values of f(xi) into the sum formula as shown above and then simplify.
I hope this clarifies the process for you!