question write a limit using summations that would equal ∫₁³ cos(5x) dx. answer attempt 1 out of 3 limₙ→∞…

Answer:

$\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}\cos\left(5\left(1+\frac{2k}{n}\right)\right)\frac{2}{n}$

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: Write the Riemann - sum

The function $f(x)=\cos(5x)$, so the Riemann - sum is $\sum_{k = 1}^{n}f(x_k)\Delta x=\sum_{k = 1}^{n}\cos\left(5\left(1+\frac{2k}{n}\right)\right)\frac{2}{n}$. Taking the limit as $n\rightarrow\infty$ gives the integral $\int_{1}^{3}\cos(5x)dx=\lim_{n\rightarrow\infty}\sum_{k = 1}^{n}\cos\left(5\left(1+\frac{2k}{n}\right)\right)\frac{2}{n}$.