NSW Y12 Maths - Advanced Integration Trapezoidal Rule

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Trapezoidal Rule Theory

\textbf{For one application} \\  \(\displaystyle \int_a^b f(x) d x \doteq \frac{1}{2}(b-a)[f(a)+f(b)]\)\\  \textbf{For \(n\) applications} \\  \(\displaystyle \int_a^b f(x) d x \doteq \dfrac{h}{2}\left[f(a)+f(b)+2\left[f\left(x_1\right)+\cdots f\left(x_{n-1}\right)\right]\right]\) \\  where \(h=\dfrac{b-a}{n}\)\\  \textbf{For 2 applications ( 3 function values)}\\  $\begin{aligned} \displaystyle \int_a^b f(x) d x & =\frac{h}{2}\left[f(a)+f(b)+2 f\left(x_1\right)\right] \\ h & =\frac{b-a}{2} \end{aligned}$\\  \textbf{For 4 applications ( 5 function values)}\\  $\begin{gathered} \displaystyle \int_a^b f(x) d x=\frac{h}{2}\left[f(a)+f(b)+2\left[f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)\right]\right] \\ h=\frac{b-a}{4} \end{gathered}$\\  \begin{multicols}{2}  \textbf{Example 1}\\ % 11005 Evaluate \(\displaystyle \int\limits_0^2 {\sqrt {1 + {x^2}} \,dx} \) using three function values\\  \textbf{Example 1 solution}\\ $\begin{aligned} \text { Area } &=\frac{1}{2}[f(0)+2 f(1)+f(2)] \\ &=\frac{1}{2}[1+2 \sqrt{2}+\sqrt{5}]=3 \cdot 03 \end{aligned}$\\  \columnbreak  \textbf{Example 2}\\ %11007 Evaluate \(\displaystyle \int\limits_1^5 {x\,{{\log }_{10}}x\,dx} \) using five function values.\\  \textbf{Example 2 solution}\\ $\begin{aligned} I &=\frac{1}{2}[f(1)+2\{f(2)+f(3)+f(4)\}+b(5)] \\ &=\frac{1}{2}\left[0+2\left\{2 \log _{10} 2+3 \log _{10} 3+4 \log _{10} 4\right\}+5 \log _{10} 5\right] \\ &=\frac{1}{2}\left[2\left\{\log _{10} 4+\log _{10} 27+\log _{10} 256\right\}+\log _{10} 3125\right] \\&= 6.2 \end{aligned}$\\  \end{multicols}

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  • Trapezoidal Rule - Video - Estimating areas using trapezoidal rule

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