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
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