f(x)=x^2-x+3 , n=6 , a=0, b=3
find LRAM, RRAM, and MRAM
for x is greater than or equal to a or less than or equal to b
You are going to do 6 rectangles between x = 0 and x = 3
for LRAM, you use the height at the left.
(I am not going to calculate them for you)
so use
f(0)=
f(.5)=
f(1)=
f(1.5)=
f(2)=
f(2.5)=
for the heights
The base of each rectangle will be 3/6 = .5
add those all up and multiply by .5
Now to do it for RRAM
use:
f(.5)=
f(1)=
......
f(3) =
then add those 6 up and multiply by .5
o do it for the middle points use
f(1/4)=
f(3/4) =
f(1 1/4) =
.......
f(2 3/4 )=
and again multiply by .5, the base of each rectangle
RRAM:20.99735189
LRAM:21.72280649
To find the Left Riemann Sum (LRAM), we divide the interval [a, b] into n subintervals of equal length. Since a = 0 and b = 3, and n = 6, each subinterval has a length of (b - a) / n = (3 - 0) / 6 = 0.5.
Now, we evaluate the function f(x) at the left endpoint of each subinterval and multiply it by the width of the subinterval.
For the LRAM, we have:
f(0) * 0.5 + f(0.5) * 0.5 + f(1) * 0.5 + f(1.5) * 0.5 + f(2) * 0.5 + f(2.5) * 0.5
Plugging in the values for f(x), we get:
(0^2 - 0 + 3) * 0.5 + (0.5^2 - 0.5 + 3) * 0.5 + (1^2 - 1 + 3) * 0.5 + (1.5^2 - 1.5 + 3) * 0.5 + (2^2 - 2 + 3) * 0.5 + (2.5^2 - 2.5 + 3) * 0.5
Simplifying the expression, we get:
(3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5
Finally, performing the calculations, we find that the Left Riemann Sum is:
1.5 + 1.5 + 1.5 + 1.5 + 1.5 + 1.5 = 9
Moving on to the Right Riemann Sum (RRAM), we evaluate the function f(x) at the right endpoint of each subinterval and multiply it by the width of the subinterval.
For the RRAM, we have:
f(0.5) * 0.5 + f(1) * 0.5 + f(1.5) * 0.5 + f(2) * 0.5 + f(2.5) * 0.5 + f(3) * 0.5
Plugging in the values for f(x), we get:
(0.5^2 - 0.5 + 3) * 0.5 + (1^2 - 1 + 3) * 0.5 + (1.5^2 - 1.5 + 3) * 0.5 + (2^2 - 2 + 3) * 0.5 + (2.5^2 - 2.5 + 3) * 0.5 + (3^2 - 3 + 3) * 0.5
Simplifying the expression, we get:
(3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5
Finally, performing the calculations, we find that the Right Riemann Sum is:
1.5 + 1.5 + 1.5 + 1.5 + 1.5 + 1.5 = 9
Lastly, we find the Midpoint Riemann Sum (MRAM). Here, we evaluate the function f(x) at the midpoint of each subinterval and multiply it by the width of the subinterval.
For the MRAM, we have:
f(0.25) * 0.5 + f(0.75) * 0.5 + f(1.25) * 0.5 + f(1.75) * 0.5 + f(2.25) * 0.5 + f(2.75) * 0.5
Plugging in the values for f(x), we get:
(0.25^2 - 0.25 + 3) * 0.5 + (0.75^2 - 0.75 + 3) * 0.5 + (1.25^2 - 1.25 + 3) * 0.5 + (1.75^2 - 1.75 + 3) * 0.5 + (2.25^2 - 2.25 + 3) * 0.5 + (2.75^2 - 2.75 + 3) * 0.5
Simplifying the expression, we get:
(3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5 + (3) * 0.5
Finally, performing the calculations, we find that the Midpoint Riemann Sum is:
1.5 + 1.5 + 1.5 + 1.5 + 1.5 + 1.5 = 9
Therefore, the Left Riemann Sum (LRAM), Right Riemann Sum (RRAM), and Midpoint Riemann Sum (MRAM) for the given function f(x)=x^2-x+3, n=6, a=0, and b=3 are all equal to 9.
To find the Left Riemann Sum (LRAM), Right Riemann Sum (RRAM), and Middle Riemann Sum (MRAM) for the given function f(x), n intervals, and interval [a, b], we need to follow these steps:
1. Determine the width of each interval (Δx):
Δx = (b - a) / n
2. Create a list of x-values for each interval:
x_i = a + i * Δx, where i represents the interval index (0 <= i <= n).
3. Calculate the corresponding y-values for each x-value:
f(x_i) = x_i^2 - x_i + 3, using the given function f(x).
4. Compute the sum of the left endpoint y-values (LRAM):
LRAM = Δx * (f(x_0) + f(x_1) + ... + f(x_n-1))
5. Compute the sum of the right endpoint y-values (RRAM):
RRAM = Δx * (f(x_1) + f(x_2) + ... + f(x_n))
6. Compute the sum of the midpoint y-values (MRAM):
MRAM = Δx * (f(x_0+Δx/2) + f(x_1+Δx/2) + ... + f(x_n-1+Δx/2))
Now, let's calculate the LRAM, RRAM, and MRAM for f(x) = x^2 - x + 3, with n = 6, a = 0, and b = 3:
1. Calculate Δx:
Δx = (3 - 0) / 6
Δx = 0.5
2. Calculate the x-values for each interval:
x_0 = 0
x_1 = 0 + 0.5 = 0.5
x_2 = 0 + 2 * 0.5 = 1
x_3 = 0 + 3 * 0.5 = 1.5
x_4 = 0 + 4 * 0.5 = 2
x_5 = 0 + 5 * 0.5 = 2.5
x_6 = 0 + 6 * 0.5 = 3
3. Calculate the y-values for each x-value:
f(x_0) = (0)^2 - 0 + 3 = 3
f(x_1) = (0.5)^2 - 0.5 + 3 = 3.75
f(x_2) = (1)^2 - 1 + 3 = 3
f(x_3) = (1.5)^2 - 1.5 + 3 = 3.75
f(x_4) = (2)^2 - 2 + 3 = 5
f(x_5) = (2.5)^2 - 2.5 + 3 = 6.25
f(x_6) = (3)^2 - 3 + 3 = 6
4. Calculate LRAM:
LRAM = 0.5 * (f(x_0) + f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5))
= 0.5 * (3 + 3.75 + 3 + 3.75 + 5 + 6.25)
= 9.375
5. Calculate RRAM:
RRAM = 0.5 * (f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5) + f(x_6))
= 0.5 * (3.75 + 3 + 3.75 + 5 + 6.25 + 6)
= 12.5
6. Calculate MRAM:
MRAM = 0.5 * (f(x_0 + 0.5/2) + f(x_1 + 0.5/2) + f(x_2 + 0.5/2) + f(x_3 + 0.5/2) + f(x_4 + 0.5/2) + f(x_5 + 0.5/2))
= 0.5 * (3 + 3.375 + 3.75 + 4.125 + 5.375 + 5.75)
= 11.125
Therefore, the LRAM is 9.375, RRAM is 12.5, and MRAM is 11.125 for the given function f(x), n = 6, a = 0, and b = 3.