question write a limit using summations that would equal ∫₁³ e^(6 - x) dx. answer attempt 1 out of 3 lim n→∞…

question write a limit using summations that would equal ∫₁³ e^(6 - x) dx. answer attempt 1 out of 3 lim n→∞ ∑ₖ = 1ⁿ

question write a limit using summations that would equal ∫₁³ e^(6 - x) dx. answer attempt 1 out of 3 lim n→∞ ∑ₖ = 1ⁿ

Answer

Answer:

$\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}e^{6-(1+\frac{2k}{n})}\cdot\frac{2}{n}$

Explanation:

Step1: Find $\Delta x$

For the integral $\int_{a}^{b}f(x)dx$ with $a = 1$, $b = 3$, $\Delta x=\frac{b - a}{n}=\frac{3 - 1}{n}=\frac{2}{n}$.

Step2: Find $x_k$

$x_k=a + k\Delta x=1+\frac{2k}{n}$.

Step3: Substitute into Riemann - sum formula

The Riemann - sum for $\int_{a}^{b}f(x)dx=\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}f(x_k)\Delta x$. Here $f(x)=e^{6 - x}$, so $f(x_k)=e^{6-(1+\frac{2k}{n})}$, and the limit of the sum is $\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}e^{6-(1+\frac{2k}{n})}\cdot\frac{2}{n}$.