Resources for Trapezoidal Rule
-
Questions
21
With Worked SolutionClick Here -
Video Tutorials
1
Click Here -
HSC Questions
3
With Worked SolutionClick Here
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}](/media/kazd3ird/4872.png)