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

The calculator will approximate the integral of $x^{2}$ from $0$ to $4$ with $n = 4$ subintervals using the left endpoint approximation, with steps shown.

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

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

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

$f{\left(x_{2} \right)} = f{\left(2 \right)} = 4$

$f{\left(x_{3} \right)} = f{\left(3 \right)} = 9$

Finally, just sum up the above values and multiply by $\Delta x = 1$: $1 \left(0 + 1 + 4 + 9\right) = 14$.

$\int\limits_{0}^{4} x^{2}\, dx\approx 14$A