Approximate $$$\int\limits_{0}^{4} x^{2} \ln\left(x\right)\, dx$$$ with $$$n = 4$$$ using the left endpoint approximation

Approximate the integral $$$\int\limits_{0}^{4} x^{2} \ln\left(x\right)\, dx$$$ with $$$n = 4$$$ 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 $$$f{\left(x \right)} = x^{2} \ln\left(x\right)$$$, $$$a = 0$$$, $$$b = 4$$$, and $$$n = 4$$$.

Therefore, $$$\Delta x = \frac{4 - 0}{4} = 1$$$.

Divide the interval $$$\left[0, 4\right]$$$ into $$$n = 4$$$ subintervals of the length $$$\Delta x = 1$$$ with the following endpoints: $$$a = 0$$$, $$$1$$$, $$$2$$$, $$$3$$$, $$$4 = b$$$.

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

There is an endpoint that doesn't belong to the domain of the function. Thus, the integral cannot be approximated.

The integral cannot be approximated.