Upper and Lower Sums (a) Prove that the following equation holds for all n Є N: 1 k³ = ¼n²(n + 1)². Σ_163 = k=1 (b) Use the equidistant partition sequence Zn = (x0,x1,...,xn) of the interval [0,6] with x kb n and determine for f(x) = x³ the upper and lower sums 5(Zn, f), S(Zn, f) depending on n and b by utilizing (a). Then justify why f is integrable over [0,6] and show that: So f(x) dx = b².

Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Upper and Lower Sums
(a) Prove that the following equation holds for all n Є N:
1 k³ = ¼n²(n + 1)².
Σ_163 =
k=1
(b) Use the equidistant partition sequence Zn = (x0,x1,...,xn) of the interval [0,6] with x
kb
n
and determine for f(x) = x³ the upper and lower sums 5(Zn, f), S(Zn, f) depending on n and
b by utilizing (a).
Then justify why f is integrable over [0,6] and show that:
So f(x) dx = b².
Transcribed Image Text:Upper and Lower Sums (a) Prove that the following equation holds for all n Є N: 1 k³ = ¼n²(n + 1)². Σ_163 = k=1 (b) Use the equidistant partition sequence Zn = (x0,x1,...,xn) of the interval [0,6] with x kb n and determine for f(x) = x³ the upper and lower sums 5(Zn, f), S(Zn, f) depending on n and b by utilizing (a). Then justify why f is integrable over [0,6] and show that: So f(x) dx = b².
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