Question
Evaluate the Riemann sum for f(r) = 5 – r2, 0 ≤ r ≤ 2 with four sub-intervals, taking the sample points to be right endpoints.
a.
8.25
b.
8.75
c.
7.25
d.
7.75
e.
6.25
a.
8.25
b.
8.75
c.
7.25
d.
7.75
e.
6.25
Answers
GPT 3.5
To evaluate the Riemann sum for this function with four sub-intervals using right endpoints, we need to calculate the width of each sub-interval (∆r) and then evaluate the function at the right endpoint of each interval.
∆r = (2-0)/4 = 0.5
The right endpoints are: 0.5, 1.0, 1.5, 2.0
Now we calculate the function values at these points:
f(0.5) = 5 - (0.5)^2 = 5 - 0.25 = 4.75
f(1.0) = 5 - 1^2 = 5 - 1 = 4
f(1.5) = 5 - (1.5)^2 = 5 - 2.25 = 2.75
f(2.0) = 5 - 2^2 = 5 - 4 = 1
Now we calculate the Riemann sum:
Riemann sum = ∆r * (f(0.5) + f(1.0) + f(1.5) + f(2.0))
Riemann sum = 0.5 * (4.75 + 4 + 2.75 + 1)
Riemann sum = 0.5 * 12.5
Riemann sum = 6.25
Therefore, the Riemann sum for the given function with four sub-intervals using right endpoints is 6.25. Therefore, the correct answer is:
e. 6.25
∆r = (2-0)/4 = 0.5
The right endpoints are: 0.5, 1.0, 1.5, 2.0
Now we calculate the function values at these points:
f(0.5) = 5 - (0.5)^2 = 5 - 0.25 = 4.75
f(1.0) = 5 - 1^2 = 5 - 1 = 4
f(1.5) = 5 - (1.5)^2 = 5 - 2.25 = 2.75
f(2.0) = 5 - 2^2 = 5 - 4 = 1
Now we calculate the Riemann sum:
Riemann sum = ∆r * (f(0.5) + f(1.0) + f(1.5) + f(2.0))
Riemann sum = 0.5 * (4.75 + 4 + 2.75 + 1)
Riemann sum = 0.5 * 12.5
Riemann sum = 6.25
Therefore, the Riemann sum for the given function with four sub-intervals using right endpoints is 6.25. Therefore, the correct answer is:
e. 6.25