Approximate $$$\int\limits_{1}^{5} \ln\left(x\right)\, dx$$$ with $$$n = 6$$$ using the left endpoint approximation

Approximate the integral $$$\int\limits_{1}^{5} \ln\left(x\right)\, dx$$$ with $$$n = 6$$$ 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 $$$f{\left(x \right)} = \ln\left(x\right)$$$, $$$a = 1$$$, $$$b = 5$$$, and $$$n = 6$$$.

Therefore, $$$\Delta x = \frac{5 - 1}{6} = \frac{2}{3}$$$.

Divide the interval $$$\left[1, 5\right]$$$ into $$$n = 6$$$ subintervals of the length $$$\Delta x = \frac{2}{3}$$$ with the following endpoints: $$$a = 1$$$, $$$\frac{5}{3}$$$, $$$\frac{7}{3}$$$, $$$3$$$, $$$\frac{11}{3}$$$, $$$\frac{13}{3}$$$, $$$5 = b$$$.

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

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

$$$f{\left(x_{1} \right)} = f{\left(\frac{5}{3} \right)} = \ln\left(\frac{5}{3}\right)\approx 0.510825623765991$$$

$$$f{\left(x_{2} \right)} = f{\left(\frac{7}{3} \right)} = \ln\left(\frac{7}{3}\right)\approx 0.847297860387204$$$

$$$f{\left(x_{3} \right)} = f{\left(3 \right)} = \ln\left(3\right)\approx 1.09861228866811$$$

$$$f{\left(x_{4} \right)} = f{\left(\frac{11}{3} \right)} = \ln\left(\frac{11}{3}\right)\approx 1.299282984130261$$$

$$$f{\left(x_{5} \right)} = f{\left(\frac{13}{3} \right)} = \ln\left(\frac{13}{3}\right)\approx 1.466337068793427$$$

Finally, just sum up the above values and multiply by $$$\Delta x = \frac{2}{3}$$$: $$$\frac{2}{3} \left(0 + 0.510825623765991 + 0.847297860387204 + 1.09861228866811 + 1.299282984130261 + 1.466337068793427\right) = 3.481570550496662.$$$

$$$\int\limits_{1}^{5} \ln\left(x\right)\, dx\approx 3.481570550496662$$$A