Give a 4-term left Riemann sum approximation for the integral below.
16
⌠ 3*((x+2)^(1/2))
⌡
12
You are welcome.
To obtain a 4-term left Riemann sum approximation for the given integral, follow the steps below:
Step 1: Determine the interval width (Δx).
In this case, the interval width is given by Δx = (b - a) / n, where a is the lower limit (12 in this case), b is the upper limit (16 in this case), and n is the number of subintervals.
Step 2: Calculate the left endpoints of each subinterval.
To calculate the left endpoints, start with the lower limit (a) and add the interval width (Δx). Repeat this process until you reach the upper limit (b).
In this case, we have 4 subintervals, so we need to calculate the left endpoints of each:
Left endpoint for the 1st subinterval: a = 12
Left endpoint for the 2nd subinterval: a + Δx = 12 + Δx
Left endpoint for the 3rd subinterval: a + 2Δx = 12 + 2Δx
Left endpoint for the 4th subinterval: a + 3Δx = 12 + 3Δx
Step 3: Evaluate the function at each left endpoint.
Substitute each left endpoint into the function (3*((x+2)^(1/2))) and evaluate it.
Left endpoint for the 1st subinterval: f(a) = 3*((12+2)^(1/2))
Left endpoint for the 2nd subinterval: f(a + Δx) = 3*((12+Δx+2)^(1/2))
Left endpoint for the 3rd subinterval: f(a + 2Δx) = 3*((12+2Δx+2)^(1/2))
Left endpoint for the 4th subinterval: f(a + 3Δx) = 3*((12+3Δx+2)^(1/2))
Step 4: Calculate the sum of the evaluated function values.
Add up the evaluated function values to get the approximation of the integral.
4-Term Left Riemann Sum = Δx * [f(a) + f(a + Δx) + f(a + 2Δx) + f(a + 3Δx)]
Therefore, the 4-term left Riemann sum approximation for the given integral is:
Δx * [3*((12+2)^(1/2)) + 3*((12+Δx+2)^(1/2)) + 3*((12+2Δx+2)^(1/2)) + 3*((12+3Δx+2)^(1/2))]
You forgot dx :)
Now I just did one of these a couple of questions down. This time use rectangles with the left hand height.
Evaluate left hand heights of function at
12
13
14
15
spacing is one so multiply each left hand height by 1 and add