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

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