write a limit using summations that would equal ∫₃⁷ ln x + 5 dx. answer attempt 3 out of 3 limₙ→∞ ∑ₖ=1ⁿ
write a limit using summations that would equal ∫₃⁷ ln x + 5 dx. answer attempt 3 out of 3 limₙ→∞ ∑ₖ=1ⁿ
Answer
Answer:
$\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}\left[\ln\left(3+\frac{4k}{n}\right)+5\right]\frac{4}{n}$
Explanation:
Step1: Find the width of sub - intervals
The interval is $[a,b]=[3,7]$, so $\Delta x=\frac{b - a}{n}=\frac{7 - 3}{n}=\frac{4}{n}$.
Step2: Find the sample points
Let $x_k=a + k\Delta x=3+\frac{4k}{n}$ for $k = 1,2,\cdots,n$.
Step3: Write the Riemann sum
The function is $f(x)=\ln x+5$. The Riemann sum is $\sum_{k = 1}^{n}f(x_k)\Delta x=\sum_{k = 1}^{n}\left[\ln\left(3+\frac{4k}{n}\right)+5\right]\frac{4}{n}$.
Step4: Take the limit as $n\rightarrow\infty$
The definite integral $\int_{3}^{7}(\ln x + 5)dx=\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}\left[\ln\left(3+\frac{4k}{n}\right)+5\right]\frac{4}{n}$.