Consider the function f(x) = x^2 on [0,1].
What is the smallest and the largest Riemann sum on n equal subintervals?
What is the smallest and the largest Riemann sum on n equal subintervals?
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Δx = (1 - 0)/n = 1/n.
Since f(x) = x^2 is increasing for non-negative x, f will always take its biggest value on a right endpoint and its smallest value on a left endpoint of an interval [a, b], with a ≥ 0.
So, the smallest Riemann sum equals
Σ(k = 1 to n) f(0 + (k-1)Δx) Δx, via left endpoints
= Σ(k = 1 to n) ((k-1)/n)^2 * (1/n).
The largest Riemann sum equals
Σ(k = 1 to n) f(0 + kΔx) Δx, via right endpoints
= Σ(k = 1 to n) (k/n)^2 * (1/n).
I hope this helps!
Since f(x) = x^2 is increasing for non-negative x, f will always take its biggest value on a right endpoint and its smallest value on a left endpoint of an interval [a, b], with a ≥ 0.
So, the smallest Riemann sum equals
Σ(k = 1 to n) f(0 + (k-1)Δx) Δx, via left endpoints
= Σ(k = 1 to n) ((k-1)/n)^2 * (1/n).
The largest Riemann sum equals
Σ(k = 1 to n) f(0 + kΔx) Δx, via right endpoints
= Σ(k = 1 to n) (k/n)^2 * (1/n).
I hope this helps!