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Reimann Sums and limit question?
what are the steps to how you should go about this question? Can you show me your steps?
1 Answer
- jibzLv 62 years agoFavorite Answer
(a) We are integrating f(x) = x⁵ from 0 to 2 so
Δx
= (2–0) / n
= 2/n,
and
x_i
= 0 + i·Δx
= 2i/n.
Therefore the integral is
lim n→∞ Σ(i=1..n) f(x_i)·Δx
= lim n→∞ Σ(i=1..n) (2i/n)⁵·2/n
= lim n→∞ (64/n⁶)Σ(i=1..n) i⁵
= [C].
(b) According to the formula
lim n→∞ (64/n⁶)Σ(i=1..n) i⁵
= lim n→∞ (64/n⁶)·n²(n+1)²(2n²+2n–1) / 12
= lim n→∞ (64/n⁶)·(2n⁶ + [lower powers of n]) / 12
= (32/3)·lim n→∞ (n⁶/n⁶ + [negative powers of n])
= (32/3)·(1 + 0)
= 32/3.
