# Approximate $\int\limits_{0}^{2} e^{- 5 x^{2}}\, dx$ with $n = 4$ using the right endpoint approximation

The calculator will approximate the integral of $e^{- 5 x^{2}}$ from $0$ to $2$ with $n = 4$ subintervals using the right endpoint approximation, with steps shown.

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Approximate the integral $\int\limits_{0}^{2} e^{- 5 x^{2}}\, dx$ with $n = 4$ using the right endpoint approximation.

### Solution

The right Riemann sum (also known as the right endpoint approximation) uses the right endpoint of a subinterval for computing the height of the approximating rectangle:

$\int\limits_{a}^{b} f{\left(x \right)}\, dx\approx \Delta x \left(f{\left(x_{1} \right)} + f{\left(x_{2} \right)} + f{\left(x_{3} \right)}+\dots+f{\left(x_{n-1} \right)} + f{\left(x_{n} \right)}\right)$

where $\Delta x = \frac{b - a}{n}$.

We have that $f{\left(x \right)} = e^{- 5 x^{2}}$, $a = 0$, $b = 2$, and $n = 4$.

Therefore, $\Delta x = \frac{2 - 0}{4} = \frac{1}{2}$.

Divide the interval $\left[0, 2\right]$ into $n = 4$ subintervals of the length $\Delta x = \frac{1}{2}$ with the following endpoints: $a = 0$, $\frac{1}{2}$, $1$, $\frac{3}{2}$, $2 = b$.

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

$f{\left(x_{1} \right)} = f{\left(\frac{1}{2} \right)} = e^{- \frac{5}{4}}\approx 0.28650479686019$

$f{\left(x_{2} \right)} = f{\left(1 \right)} = e^{-5}\approx 0.006737946999085$

$f{\left(x_{3} \right)} = f{\left(\frac{3}{2} \right)} = e^{- \frac{45}{4}}\approx 0.000013007297654$

$f{\left(x_{4} \right)} = f{\left(2 \right)} = e^{-20}\approx 2.061154 \cdot 10^{-9}$

Finally, just sum up the above values and multiply by $\Delta x = \frac{1}{2}$: $\frac{1}{2} \left(0.28650479686019 + 0.006737946999085 + 0.000013007297654 + 2.061154 \cdot 10^{-9}\right) = 0.146627876609041.$

$\int\limits_{0}^{2} e^{- 5 x^{2}}\, dx\approx 0.146627876609041$A