Find the Riemann sum associated with
f(x)=4x^2+2, n=3 and the partition
xsub0=-2
xsub1=1
xsub2=2
xsub3=4
of [-2,4]
(a) when xsubk is the right end-point of [xsubk-1, xsubk] .
(b) when when xsubk is the mid-point of [xsubk-1, xsubk]
please help. i thought fora the anser was 216 but that's not right and i keep getting that answer so i must be doing something wrong. i thought u have to take the definate integral of the equation given from -2 to 4 and then multiply that by two..please tell me what i am doing wrong and what numbers to use for b...thank you!!
To find the Riemann sum associated with the given function, we need to divide the interval [-2, 4] into subintervals and evaluate the function at specific points within each subinterval.
(a) When xsubk is the right endpoint of [xsubk-1, xsubk]:
Step 1: Calculate the width of each subinterval.
The width of each subinterval is given by Δx = (b - a) / n, where b is the upper endpoint of the interval, a is the lower endpoint, and n is the number of subintervals. In this case, b = 4, a = -2, and n = 3. Therefore, Δx = (4 - (-2)) / 3 = 6 / 3 = 2.
Step 2: Determine the right endpoints.
Since xsubk is the right endpoint of each subinterval, we have:
xsub0 = -2,
xsub1 = -2 + 2 = 0,
xsub2 = 0 + 2 = 2,
xsub3 = 2 + 2 = 4.
Step 3: Evaluate the function at each right endpoint.
Evaluate f(x) = 4x^2 + 2 at xsub0, xsub1, xsub2, and xsub3:
f(xsub0) = 4(-2)^2 + 2 = 16 + 2 = 18,
f(xsub1) = 4(0)^2 + 2 = 0 + 2 = 2,
f(xsub2) = 4(2)^2 + 2 = 16 + 2 = 18,
f(xsub3) = 4(4)^2 + 2 = 64 + 2 = 66.
Step 4: Calculate the Riemann sum.
The Riemann sum, with xsubk as the right endpoint, is given by:
R = f(xsub0) * Δx + f(xsub1) * Δx + f(xsub2) * Δx + f(xsub3) * Δx.
Plugging in the previously calculated values:
R = 18 * 2 + 2 * 2 + 18 * 2 + 66 * 2
R = 36 + 4 + 36 + 132
R = 208.
Therefore, the Riemann sum associated with f(x) = 4x^2 + 2, n = 3, and xsubk as the right endpoint is 208.
(b) When xsubk is the midpoint of [xsubk-1, xsubk]:
To find the Riemann sum with xsubk as the midpoint, we follow the same steps as above, but calculate the midpoints of each subinterval.
Step 1: Calculate the width of each subinterval (Δx) as before.
Step 2: Determine the midpoints.
The midpoint of each subinterval is given by (xsubk-1 + xsubk) / 2.
For this problem, the midpoints are:
(xsub0 + xsub1) / 2 = (-2 + 0) / 2 = -1,
(xsub1 + xsub2) / 2 = (0 + 2) / 2 = 1,
(xsub2 + xsub3) / 2 = (2 + 4) / 2 = 3.
Step 3: Evaluate the function at each midpoint.
Evaluate f(x) = 4x^2 + 2 at each midpoint:
f(-1) = 4(-1)^2 + 2 = 4 + 2 = 6,
f(1) = 4(1)^2 + 2 = 4 + 2 = 6,
f(3) = 4(3)^2 + 2 = 36 + 2 = 38.
Step 4: Calculate the Riemann sum.
Using the midpoint values, we have:
R = f(-1) * Δx + f(1) * Δx + f(3) * Δx.
Plugging in the previously calculated values:
R = 6 * 2 + 6 * 2 + 38 * 2
R = 12 + 12 + 76
R = 100.
Therefore, the Riemann sum associated with f(x) = 4x^2 + 2, n = 3, and xsubk as the midpoint is 100.
Make sure to double-check your calculations and follow the steps accurately to avoid any errors.