The Trapezium Rule

\[\int ^b_{a}y \, dx =\frac{1}{2}h(y_{0}+2(y_{1}+y_{2}+y_{3}+y_{n-1})+y_{n})$$ $$h=\frac{b-a}{n}\]

Example

Line is $y=\sec x$ between $0$ and $\frac{\pi}{3}$

| $x$ | $0$ | $\frac{\pi}{12}$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ |
|---|---|-------|-------|-------|------|
| $y$ | $1$ | $1.035$ | $1.155$ | $1.414$ | $2$ |

- Using trapezium rule:

$\(\int _{a}^by \, dx = \frac{1}{2}\times \frac{\pi}{12}[1+2(1.035+1.155+1.414)+2]=\frac{\pi}{24}[10.208]=1.336\dots=1.34\text{ to 2dp}$\)

Ex 11I Q1

Line is $y=\sqrt{ 1+\tan x }$ between $0$ and $\frac{\pi}{3}$

| $x$ | $0$ | $\frac{\pi}{12}$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ |
|---|---|-------|-------|-------|------|
| $y$ | $1$ | $1.1233$ | $1.2559$ | $1.3362$ | $1.6529$ |

- Using trapezium rule and values for $0$, $\frac{\pi}{6}$ and $\frac{\pi}{3}$:

$\(\int _{a}^by \, dx = \frac{1}{2}\times \frac{\pi}{6}(1+2(1.2559)+1.6529)=\frac{\pi}{12}\times5.1647=1.3521\dots=1.352\text{ to 4sf}$\)

- Using trapezium rule and values for $0$, $\frac{\pi}{12}$, $\frac{\pi}{6}$, $\frac{\pi}{4}$ and $\frac{\pi}{3}$:

$\(\int _{a}^by \, dx = \frac{1}{2}\times \frac{\pi}{12}(1+2(1.1233+1.2559+1.3362)+1.6529)=\frac{\pi}{24}\times_{1}0.0837=1.31995\dots=1.320\text{ to 4sf}$\)

Ex 11I Q2

Line is $y=\cos\frac{5\theta}{2}$ between $-\frac{\pi}{5}$ and $\frac{\pi}{5}$

| $x$ | $-\frac{\pi}{5}$ | $-\frac{\pi}{10}$ | $0$ | $\frac{\pi}{10}$ | $\frac{\pi}{5}$ |
|---|---|-------|-------|-------|------|
| $y$ | $0$ | $\frac{\sqrt{ 2 }}{2}$ | $1$ | $\frac{\sqrt{ 2 }}{2}$ | $0$ |

- Using trapezium rule and all values in the table:

$\(\int _{a}^by \, dx = \frac{1}{2}\times\frac{\pi}{10}\left( 2\left( \frac{\sqrt{ 2 }}{2}+1+\frac{\sqrt{ 2 }}{2} \right) \right)=\frac{\pi}{10}(1+\sqrt{ 2 })=0.75845\dots=0.758\text{ to 3dp}$\)

- Trapezium rule produces an underestimate due to trapezium lines going under the curve

$\(\int ^{\frac{\pi}{5}}_{-\frac{\pi}{5}} \cos\frac{5\theta}{2}\, dx =\left[ \frac{2}{5}\sin\frac{5\theta}{2} \right]^{\frac{\pi}{5}}_{-\frac{\pi}{5}}=\frac{2}{5}--\frac{2}{5}=\frac{4}{5}=0.8$\)

$5.25\% \text{ error}$

The Trapezium Rule

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