Which of the definite integrals is equivalent to the following limit?
$$
\lim _{n \rightarrow \infty} \sum_{i=1}^{n} 4 \cdot \frac{5}{n}
$$

Choose 1 answer:
(A) $\int_{0}^{5} 4 d x$
(B) $\int_{0}^{5} 4 x d x$
(c) $\int_{0}^{4} 5 x d x$
(D) $\int_{0}^{4} 5 d x$

Question

Which of the definite integrals is equivalent to the following limit?
$$
\lim _{n \rightarrow \infty} \sum_{i=1}^{n} 4 \cdot \frac{5}{n}
$$

Choose 1 answer:
(A) $\int_{0}^{5} 4 d x$
(B) $\int_{0}^{5} 4 x d x$
(c) $\int_{0}^{4} 5 x d x$
(D) $\int_{0}^{4} 5 d x$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Consider the given limit as the definition of a definite integral: $\lim_{n o \infty} \sum_{i=1}^{n} 4 \cdot \frac{5}{n}$. ‖ ‖ step 2 ⋮ Recognize that $4 \cdot \frac{5}{n}$ is the function value times the width of each subinterval $\frac{5}{n}$ in the Riemann sum for the integral. ‖ ‖ step 3 ⋮ The function being integrated is constant, $f(x) = 4$, over the interval [0, 5]. ‖ ‖ step 4 ⋮ The limit represents the definite integral of $f(x) = 4$ from 0 to 5, which is $\int_{0}^{5} 4 dx$. ‖ ***A*** ∻Key Concept∻ ⚹Riemann Sum to Definite Integral⚹ ∻Explanation∻ ⚹The limit of the sum as $n$ approaches infinity represents the definite integral of a constant function over a specified interval.⚹◊What is the equivalent definite integral of the given limit in terms of $n$?⍭ Generate me a similar question◊

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