Editing Trapezoidal rule (section)

==Example==
The following integral is given: 
<math display="block"> \int_{0.1}^{1.3}{5xe^{- 2x}{dx}} </math> 
{{ordered list| list-style-type = lower-alpha
| Use the composite trapezoidal rule to estimate the value of this integral. Use three segments. 
 
| Find the true error <math display="inline"> E_{t} </math> for part (a). 
 
| Find the absolute relative true error <math display="inline"> \left| \varepsilon_{t} \right| </math> for part (a). 
}}
'''Solution''' 
{{ordered list| list-style-type = lower-alpha
| The solution using the composite trapezoidal rule with 3 segments is applied as follows. 
<math display="block"> \int_{a}^{b}{f(x){dx}} \approx \frac{b - a}{2n}\left\lbrack f(a) + 2\sum_{i = 1}^{n - 1}{f(a + {ih})} + f(b) \right\rbrack </math> 
 
<math display="block">\begin{align}
n &= 3 \\
a &= 0.1 \\
b &= 1.3 \\
h &= \frac{b - a}{n}
= \frac{1.3 - 0.1}{3}
= 0.4
\end{align} </math> 
 
Using the composite trapezoidal rule formula
<math display="block"> \begin{align} \int_a^b {f(x){dx}} \approx \frac{b - a}{2n} \left\lbrack f(a) + 2\left\{ \sum_{i = 1}^{n - 1}{f(a + {ih})} \right\} + f(b) \right\rbrack\;\;\;\;\;\;\;\;\;\;\;\; (3) \end{align} </math> 
 
<math display="block"> \begin{align}
 I &\approx \frac{1.3 - 0.1}{6}\left\lbrack f(0.1) + 2\sum_{i = 1}^{3 - 1}{f(0.1 + 0.4i)} + f(1.3) \right\rbrack\\
 I &\approx \frac{1.3 - 0.1}{6}\left\lbrack f(0.1) + 2\sum_{i = 1}^{2}{f(0.1 + 0.4i)} + f(1.3) \right\rbrack\\
&= 0.2\lbrack f(0.1) + 2f(0.5) + 2f(0.9) + f(1.3)\rbrack\\
 &=  0.2[5 \times 0.1 \times e^{- 2(0.1)}+2(5 \times 0.5 \times e^{- 2(0.5)})+2(5 \times 0.9 \times e^{- 2(0.9)}) + 5 \times 1.3 \times e^{- 2(1.3)}\rbrack\\
 &= 0.84385
 \end{align} </math> 
 
| The exact value of the above integral can be found by integration by parts and is 
<math display="block"> \int_{0.1}^{1.3} 5xe^{- 2x}{dx} = 0.89387 </math> 
 
So the true error is
<math display="block"> \begin{align}
 E_{t} &= \text{True Value} - \text{Approximate Value}\\
 &= 0.89387 - 0.84385\\
 &= 0.05002
\end{align} </math> 
 
| The absolute relative true error is
<math display="block"> \displaystyle \begin{align}\left| \varepsilon_{t} \right| &= \left| \frac{\text{True Error}}{\text{True Value}} \right| \times 100\%\\
 &= \left| \frac{0.05002}{0.89387} \right| \times 100\%\\
 &= 5.5959\%
\end{align} </math>
}}