Which of the following limits is equal to ∫31sin(x3+2)ⅆx ?
To determine which of the following limits is equal to the integral ∫31sin(x^3+2)dx, we can compare the given options with the properties of definite integrals
To determine which of the following limits is equal to the integral ∫31sin(x^3+2)dx, we can compare the given options with the properties of definite integrals.
Option 1: lim(n→∞) ∑(k=0 to n) sin(3 + 3k/n) * 6/n
This option represents a Riemann sum approximation of the integral. However, the given sum does not correspond to the same function nor interval as the original integral. Therefore, we can exclude this option.
Option 2: lim(h→0) ∑(k=1 to ∞) sin^3(3 + kh) * h
Similarly to the previous option, this is also a Riemann sum approximation. The limits within the sum do not match the original integral, so this option can be excluded as well.
Option 3: lim(n→∞) ∑(k=1 to n) sin(3 + 6k/n) * 3/n
Unlike the previous options, this Riemann sum represents the original integral more closely. By rewriting it, we can see that it approaches ∫31sin(x^3+2)dx as n approaches infinity. Therefore, this option is the correct answer.
In summary, the limit that is equal to ∫31sin(x^3+2)dx is lim(n→∞) ∑(k=1 to n) sin(3 + 6k/n) * 3/n (Option 3).
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