Approximate $$$\int\limits_{1}^{3} \frac{\sin{\left(x \right)}}{x}\, dx$$$ with $$$n = 2$$$ using the left endpoint approximation

Approximate the integral $$$\int\limits_{1}^{3} \frac{\sin{\left(x \right)}}{x}\, dx$$$ with $$$n = 2$$$ using the left endpoint approximation.

The left Riemann sum (also known as the left endpoint approximation) uses the left endpoint of a subinterval for computing the height of the approximating rectangle:

$$$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \Delta x \left(f{\left(x_{0} \right)} + f{\left(x_{1} \right)} + f{\left(x_{2} \right)}+\dots+f{\left(x_{n-2} \right)} + f{\left(x_{n-1} \right)}\right)$$$

where $$$\Delta x = \frac{b - a}{n}$$$.

We have that $$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x}$$$, $$$a = 1$$$, $$$b = 3$$$, and $$$n = 2$$$.

Therefore, $$$\Delta x = \frac{3 - 1}{2} = 1$$$.

Divide the interval $$$\left[1, 3\right]$$$ into $$$n = 2$$$ subintervals of the length $$$\Delta x = 1$$$ with the following endpoints: $$$a = 1$$$, $$$2$$$, $$$3 = b$$$.

Now, just evaluate the function at the left endpoints of the subintervals.

$$$f{\left(x_{0} \right)} = f{\left(1 \right)} = \sin{\left(1 \right)}\approx 0.841470984807897$$$

$$$f{\left(x_{1} \right)} = f{\left(2 \right)} = \frac{\sin{\left(2 \right)}}{2}\approx 0.454648713412841$$$

Finally, just sum up the above values and multiply by $$$\Delta x = 1$$$: $$$1 \left(0.841470984807897 + 0.454648713412841\right) = 1.296119698220738.$$$

$$$\int\limits_{1}^{3} \frac{\sin{\left(x \right)}}{x}\, dx\approx 1.296119698220738$$$A