which of the following definite integrals is equal to lim n→∞ ∑k = 1n 10k / n (√(1 + 5k / n)) (5 / n)? a…

which of the following definite integrals is equal to lim n→∞ ∑k = 1n 10k / n (√(1 + 5k / n)) (5 / n)? a ∫1^6 10√x dx b ∫1^6 2√x dx c ∫0^5 10√1 + x dx d ∫0^5 2√1 + x dx

which of the following definite integrals is equal to lim n→∞ ∑k = 1n 10k / n (√(1 + 5k / n)) (5 / n)? a ∫1^6 10√x dx b ∫1^6 2√x dx c ∫0^5 10√1 + x dx d ∫0^5 2√1 + x dx

Answer

Explanation:

Step1: Recall the definition of definite - integral as a limit of Riemann sum

The definite integral $\int_{a}^{b}f(x)dx=\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}f(x_k)\Delta x$, where $\Delta x=\frac{b - a}{n}$ and $x_k=a + k\Delta x$.

Step2: Identify $\Delta x$ and $x_k$ from the given limit

In the sum $\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}\frac{10k}{n}\sqrt{1+\frac{5k}{n}}\left(\frac{5}{n}\right)$, we have $\Delta x=\frac{5}{n}$. Then, since $\Delta x=\frac{b - a}{n}$, we get $b - a = 5$. Also, $x_k=\frac{5k}{n}$, which implies $a = 0$ and $b=5$. And $f(x_k)=\frac{10k}{n}\sqrt{1+\frac{5k}{n}}$. If we let $x=\frac{5k}{n}$, then $f(x)=2x\sqrt{1 + x}$.

Step3: Write the definite - integral

The definite integral corresponding to the given limit is $\int_{0}^{5}2x\sqrt{1 + x}dx$.

Answer:

D. $\int_{0}^{5}2x\sqrt{1 + x}dx$