Answer
$$
\int_{0}^{2}\frac{e^{x}}{1+x^{2}}dx, \quad n=10
$$
(a) The approximation of the given integral by using the Trapezoidal Rule is $\approx 2.660833 $
(b) The approximation of the given integral by using the midpoints Rule is $\approx 2.664377 $
(c) The approximation of the given integral by using Simpson’s Rule is $\approx 2.663244 $
Work Step by Step
$$
\int_{0}^{2}\frac{e^{x}}{1+x^{2}}dx, \quad n=10
$$
(a) Use the Trapezoidal Rule to approximate the given integral with the
specified value of n.
With $ n =10, a = 0$, and $b = 2$ we have
$$
\Delta x=\frac{b-a}{n}=\frac{2-0}{10}=\frac{1}{5}
$$ and so the Trapezoidal Rule gives:
$$
\begin{aligned} \int_{0}^{2}\frac{e^{x}}{1+x^{2}}dx &= \frac{\Delta x}{2}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]\\
& \approx T_{10} \\
& =\frac{1}{5 \cdot 2}[f(0)+2 f(0.2)+2 f(0.4)+2 f(0.6)+\\
& \quad\quad+2 f(0.8)+2 f(1) +2 f(1.2)+2 f(1.4)+\\ &
\quad\quad+2 f(1.6)+2 f(1.8)+f(2)] \\
& \approx 2.660833
\end{aligned}
$$
(b) The midpoints Rule to approximate the given integral with the
specified value of $ n=10$ gives :
$$
\begin{aligned}\int_{0}^{2}\frac{e^{x}}{1+x^{2}}dx &=\Delta x\left[f\left(\overline{x}_{1}\right)+f\left(\overline{x}_{2}\right)+\cdots+f\left(\overline{x}_{n}\right)\right] \\
& = M_{10} \\
& \approx
\frac{1}{5}[f(0.1)+f(0.3)+f(0.5)+f(0.7)+f(0.9)+\\
&\quad\quad +f(1.1)+f(1.3)+f(1.5)+f(1.7)+f(1.9)] \\
& \approx 2.664377 \end{aligned}
$$
(c) Simpson’s Rule to approximate the given integral with the
specified value of $ n=10$ gives :
$$
\begin{aligned} \int_{0}^{2}\frac{e^{x}}{1+x^{2}}dx & = \frac{\Delta x}{3}\left[f\left(x_{0}\right)\right. +4 f\left(x_{1}\right)+2 f\left(x_{2}\right)+4 f\left(x_{3}\right)+\cdots \\ & \quad \left.+2 f\left(x_{n-2}\right)+4 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right] \\
&= S_{10}\\
& \approx \frac{1}{5 \cdot 3}[f(0)+4 f(0.2)+2 f(0.4)+4 f(0.6)+\\
&\quad\quad + 2 f(0.8) +4 f(1)+2 f(1.2)+4 f(1.4)+\\
& \quad\quad+2 f(1.6)+4 f(1.8)+f(2)] \\
& \approx 2.663244 \end{aligned}
$$
