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

The calculator will approximate the integral of $4 x^{3} e^{- x}$ from $-1$ to $1$ with $n = 7$ subintervals using the left endpoint approximation, with steps shown.

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### Your Input

Approximate the integral $\int\limits_{-1}^{1} 4 x^{3} e^{- x}\, dx$ with $n = 7$ 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 $f{\left(x \right)} = 4 x^{3} e^{- x}$, $a = -1$, $b = 1$, 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 $\Delta x = \frac{2}{7}$: $\frac{2}{7} \left(-10.873127313836181 - 2.977736254032277 - 0.483343454809221 - 0.013452653001692 + 0.010109363262393 + 0.205117837356717 + 0.713617579529086\right) = -3.833947113008907.$

### Answer

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