Consider the function f(x)=-((x^2)/2)-9.
In this problem you will calculate integrate from 0 to 3 of ((-x^2)/2)-9)dx by using the definition
integrate from a to b of (f(x))dx= lim as n approaches infinity of sum_(i=1)^n of (f(x_i))(delta x)
The summation inside the brackets is R_n which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub-interval.
Calculate R_n for f(x)=((x^2)/2)-9 on the interval [0,3] and write your answer as a function of n without any summation signs.
To calculate R_n for the function f(x) = ((x^2)/2)-9 on the interval [0, 3], we'll follow the steps of the definition.
Step 1: Determine the interval [a, b]
The interval given is [0, 3], so a = 0 and b = 3.
Step 2: Find the width of each sub-interval Δx
The width of each sub-interval, Δx, is calculated by dividing the length of the interval (b - a) by the number of sub-intervals (n). In this case, since the interval is [0, 3], b - a = 3 - 0 = 3. So, Δx = (b - a)/n = 3/n.
Step 3: Determine the right-hand endpoint of each sub-interval
Since the sample points are chosen to be the right-hand endpoints of each sub-interval, we'll calculate the x-values for each right-hand endpoint. Starting from the left endpoint a = 0, we add Δx repeatedly to find each right-hand endpoint.
Right-hand endpoints:
x_1 = a + Δx = 0 + 3/n
x_2 = x_1 + Δx = (0 + 3/n) + 3/n = 6/n
x_3 = x_2 + Δx = (0 + 3/n) + 3/n + 3/n = 9/n
...
x_n = a + (n-1)Δx = 0 + (n-1)(3/n) = 3(n-1)/n
Step 4: Calculate f(x_i) for each right-hand endpoint x_i
Evaluate the function f(x) = ((x^2)/2)-9 at each right-hand endpoint from Step 3.
f(x_1) = ((3/n)^2)/2 - 9 = (9/n^2)/2 - 9 = 9/(2n^2) - 9
f(x_2) = ((6/n)^2)/2 - 9 = (36/n^2)/2 - 9 = 18/n^2 - 9
f(x_3) = ((9/n)^2)/2 - 9 = (81/n^2)/2 - 9 = 81/(2n^2) - 9
...
f(x_n) = ([(3(n-1))/n]^2)/2 - 9
Step 5: Write the Riemann sum (R_n) as a function of n without any summation signs
Now, we'll write R_n as a function of n without any summation signs by summing each term (f(x_i))(Δx) for i varying from 1 to n.
R_n = f(x_1)Δx + f(x_2)Δx + f(x_3)Δx + ... + f(x_n)Δx
= [9/(2n^2) - 9] * (3/n) + [18/n^2 - 9] * (3/n) + [81/(2n^2) - 9] * (3/n) + ... + [([(3(n-1))/n]^2)/2 - 9] * (3/n)
Therefore, the Riemann sum R_n for f(x) = ((x^2)/2)-9 on the interval [0, 3] can be written as a function of n without any summation signs as:
R_n = [9/(2n^2) - 9] * (3/n) + [18/n^2 - 9] * (3/n) + [81/(2n^2) - 9] * (3/n) + ... + [([(3(n-1))/n]^2)/2 - 9] * (3/n)
To calculate R_n for f(x) = ((x^2)/2) - 9 on the interval [0, 3], we first need to determine the right-hand endpoints of each sub-interval.
We can find the length of each sub-interval by dividing the total interval length by the number of sub-intervals, which is given by (b - a)/n. In this case, a = 0, b = 3, and n represents the number of sub-intervals.
The right-hand endpoints of each sub-interval can be calculated by adding the length of each sub-interval to the left endpoint of that interval. Thus, the right-hand endpoint for the i-th sub-interval can be found using the formula: x_i = a + i * (b - a)/n.
Let's calculate R_n step by step:
1. Determine the interval length:
interval length = (b - a) = (3 - 0) = 3.
2. Calculate the length of each sub-interval:
sub-interval length = interval length / n = 3 / n.
3. Calculate the right-hand endpoints of each sub-interval:
x_i = 0 + i * (3/n), where i ranges from 1 to n.
4. Evaluate f(x_i) for each right-hand endpoint:
f(x_i) = ((x_i^2) / 2) - 9.
5. Calculate the Riemann sum:
R_n = (sub-interval length) * (f(x_1) + f(x_2) + ... + f(x_n)).
Now, let's write the function R_n without any summation signs:
R_n = (3 / n) * (((0 + (1 * (3/n))^2) / 2) - 9) + (((0 + (2 * (3/n))^2) / 2) - 9) + ... + (((0 + (n * (3/n))^2) / 2) - 9).
Simplifying further:
R_n = (3 / n) * ((1^2 * (9/n^2)) / 2 - 9) + ((2^2 * (9/n^2)) / 2 - 9) + ... + ((n^2 * (9/n^2)) / 2 - 9).
R_n = (3 / n) * (((9/n^2) / 2) - 9) + (((36/n^2) / 2) - 9) + ... + (((n^2 * (9/n^2)) / 2) - 9).
R_n = (3 / n) * ((9/2n^2) - 9) + ((36/2n^2) - 9) + ... + (((n^2 * (9/n^2)) / 2) - 9).
Therefore, the Riemann sum R_n for f(x) = ((x^2)/2) - 9 on the interval [0, 3] can be written as a function of n without any summation signs as:
R_n = (3 / n) * ((9/2n^2) - 9) + ((36/2n^2) - 9) + ... + (((n^2 * (9/n^2)) / 2) - 9).