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

The calculator will approximate the integral of $\ln\left(x\right)$ from $1$ to $5$ with $n = 6$ subintervals using the left endpoint approximation, with steps shown.

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

### Solution

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