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

The calculator will approximate the integral of $x^{2} \ln\left(x\right)$ from $0$ to $4$ with $n = 4$ subintervals using the Riemann sum, with steps shown.

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

### 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 $f{\left(x \right)} = x^{2} \ln\left(x\right)$, $a = 0$, $b = 4$, 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.