# Approximate $\int\limits_{1}^{4} \left(x^{2} - 4 x + 2\right)\, dx$ with $n = 9$ using the left endpoint approximation

The calculator will approximate the integral of $x^{2} - 4 x + 2$ from $1$ to $4$ with $n = 9$ subintervals using the left endpoint approximation, with steps shown.

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Approximate the integral $\int\limits_{1}^{4} \left(x^{2} - 4 x + 2\right)\, dx$ with $n = 9$ 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 $f{\left(x \right)} = x^{2} - 4 x + 2$, $a = 1$, $b = 4$, and $n = 9$.

Therefore, $\Delta x = \frac{4 - 1}{9} = \frac{1}{3}$.

Divide the interval $\left[1, 4\right]$ into $n = 9$ subintervals of the length $\Delta x = \frac{1}{3}$ with the following endpoints: $a = 1$, $\frac{4}{3}$, $\frac{5}{3}$, $2$, $\frac{7}{3}$, $\frac{8}{3}$, $3$, $\frac{10}{3}$, $\frac{11}{3}$, $4 = b$.

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

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

$f{\left(x_{1} \right)} = f{\left(\frac{4}{3} \right)} = - \frac{14}{9}\approx -1.555555555555556$

$f{\left(x_{2} \right)} = f{\left(\frac{5}{3} \right)} = - \frac{17}{9}\approx -1.888888888888889$

$f{\left(x_{3} \right)} = f{\left(2 \right)} = -2$

$f{\left(x_{4} \right)} = f{\left(\frac{7}{3} \right)} = - \frac{17}{9}\approx -1.888888888888889$

$f{\left(x_{5} \right)} = f{\left(\frac{8}{3} \right)} = - \frac{14}{9}\approx -1.555555555555556$

$f{\left(x_{6} \right)} = f{\left(3 \right)} = -1$

$f{\left(x_{7} \right)} = f{\left(\frac{10}{3} \right)} = - \frac{2}{9}\approx -0.222222222222222$

$f{\left(x_{8} \right)} = f{\left(\frac{11}{3} \right)} = \frac{7}{9}\approx 0.777777777777778$

Finally, just sum up the above values and multiply by $\Delta x = \frac{1}{3}$: $\frac{1}{3} \left(-1 - 1.555555555555556 - 1.888888888888889 - 2 - 1.888888888888889 - 1.555555555555556 - 1 - 0.222222222222222 + 0.777777777777778\right) = -3.444444444444445.$

$\int\limits_{1}^{4} \left(x^{2} - 4 x + 2\right)\, dx\approx -3.444444444444445$A