Approximate $$$\int\limits_{-1}^{1} 4 x^{3} e^{- x}\, dx$$$ with $$$n = 7$$$ using the left endpoint approximation

Approximate the integral $$$\int\limits_{-1}^{1} 4 x^{3} e^{- x}\, dx$$$ with $$$n = 7$$$ 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 $$$f{\left(x \right)} = 4 x^{3} e^{- x}$$$, $$$a = -1$$$, $$$b = 1$$$, and $$$n = 7$$$.

Therefore, $$$\Delta x = \frac{1 - \left(-1\right)}{7} = \frac{2}{7}$$$.

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

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

$$$f{\left(x_{0} \right)} = f{\left(-1 \right)} = - 4 e\approx -10.873127313836181$$$

$$$f{\left(x_{1} \right)} = f{\left(- \frac{5}{7} \right)} = - \frac{500 e^{\frac{5}{7}}}{343}\approx -2.977736254032277$$$

$$$f{\left(x_{2} \right)} = f{\left(- \frac{3}{7} \right)} = - \frac{108 e^{\frac{3}{7}}}{343}\approx -0.483343454809221$$$

$$$f{\left(x_{3} \right)} = f{\left(- \frac{1}{7} \right)} = - \frac{4 e^{\frac{1}{7}}}{343}\approx -0.013452653001692$$$

$$$f{\left(x_{4} \right)} = f{\left(\frac{1}{7} \right)} = \frac{4}{343 e^{\frac{1}{7}}}\approx 0.010109363262393$$$

$$$f{\left(x_{5} \right)} = f{\left(\frac{3}{7} \right)} = \frac{108}{343 e^{\frac{3}{7}}}\approx 0.205117837356717$$$

$$$f{\left(x_{6} \right)} = f{\left(\frac{5}{7} \right)} = \frac{500}{343 e^{\frac{5}{7}}}\approx 0.713617579529086$$$

Finally, just sum up the above values and multiply by $$$\Delta x = \frac{2}{7}$$$: $$$\frac{2}{7} \left(-10.873127313836181 - 2.977736254032277 - 0.483343454809221 - 0.013452653001692 + 0.010109363262393 + 0.205117837356717 + 0.713617579529086\right) = -3.833947113008907.$$$

$$$\int\limits_{-1}^{1} 4 x^{3} e^{- x}\, dx\approx -3.833947113008907$$$A