Question Statement

The values of a certain function $f(x)$ are given in the following table:

Using the Trapezoidal rule and Simpson's $\frac{1}{3}$ rule, compute the integral ${\int }_0^{4}f(x)dx$.

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Background and Explanation

This problem requires numerical integration of tabulated data where the explicit form of $f(x)$ is unknown. When data points are equally spaced, Newton-Cotes formulas—specifically the Trapezoidal rule and Simpson's $\frac{1}{3}$ rule—provide efficient methods for approximating definite integrals by weighted summation of function values.

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Solution

Step 1: Determine Step Size and Function Values

Observe from the table that the $x$ values are equally spaced with a difference of $0.5$. Therefore, the step size is: $h=0.5$

The interval $[0,4]$ is divided into 8 subintervals with 9 data points ($x_0$ through $x_8$). The corresponding function values are:

$\begin{array}{rl}f(x_0)& =f(0)=1\\ f(x_1)& =f(0.5)=1.649\\ f(x_2)& =f(1)=2.718\\ f(x_3)& =f(1.5)=4.482\\ f(x_4)& =f(2)=7.389\\ f(x_5)& =f(2.5)=12.18\\ f(x_6)& =f(3)=20.09\\ f(x_7)& =f(3.5)=22.12\\ f(x_8)& =f(4)=45.60\end{array}$

Part (a): Trapezoidal Rule

The Trapezoidal rule formula for $n$ subintervals is: ${\int }_0^{4}f(x)dx\approx \frac{h}{2}\left[f(x_0)+2{\sum }_{i=1}^{n-1}f(x_i)+f(x_n)\right]$

Substituting $h=0.5$ and expanding the summation: ${\int }_0^{4}f(x)dx\approx \frac{0.5}{2}\left[f(x_0)+2\left(f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)+f(x_6)+f(x_7)\right)+f(x_8)\right]$

Plugging in the numerical values: $\begin{array}{rl}{\int }_0^{4}f(x)dx& \approx \frac{0.5}{2}\left[1+2(1.649+2.718+4.482+7.389+12.18+20.09+22.12)+45.60\right]\\ & \approx 0.25\left[1+2(70.628)+45.60\right]\\ & \approx 0.25\left[1+141.256+45.60\right]\\ & \approx 0.25(187.856)\\ & \approx 46.964\end{array}$

Part (b): Simpson's $\frac{1}{3}$ Rule

Since we have an even number of subintervals ($n=8$), Simpson's $\frac{1}{3}$ rule applies: ${\int }_0^{4}f(x)dx\approx \frac{h}{3}\left[f(x_0)+4{\sum }_{\text{odd }i}f(x_i)+2{\sum }_{\text{even }i}f(x_i)+f(x_8)\right]$

Expanded form: ${\int }_0^{4}f(x)dx\approx \frac{h}{3}\left[f(x_0)+4(f(x_1)+f(x_3)+f(x_5)+f(x_7))+2(f(x_2)+f(x_4)+f(x_6))+f(x_8)\right]$

Substituting the values: $\begin{array}{rl}{\int }_0^{4}f(x)dx& \approx \frac{0.5}{3}\left[1+4(1.649+4.482+12.18+22.12)+2(2.718+7.389+20.09)+45.60\right]\\ & \approx 0.167\left[1+4(40.431)+2(30.197)+45.60\right]\\ & \approx 0.167\left[1+161.724+60.394+45.60\right]\\ & \approx 0.167(268.718)\\ & \approx 44.876\end{array}$

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Key Formulas or Methods Used

• Trapezoidal Rule: ${\int }_a^{b}f(x)dx\approx \frac{h}{2}\left[y_0+2(y_1+y_2+\cdots +y_{n-1})+y_n\right]$ where $h=\frac{b-a}{n}$ and $n$ is the number of subintervals
• Simpson's $\frac{1}{3}$ Rule: ${\int }_a^{b}f(x)dx\approx \frac{h}{3}\left[y_0+4(y_1+y_3+\cdots +y_{n-1})+2(y_2+y_4+\cdots +y_{n-2})+y_n\right]$ (requires $n$ to be even)

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Summary of Steps

1. Identify parameters: Determine step size $h=0.5$ and confirm 8 subintervals (9 data points) exist
2. List function values: Extract $f(x_0)$ through $f(x_8)$ from the table
3. Apply Trapezoidal Rule: Calculate $\frac{h}{2}\times [\text{first}+\text{last}+2\times (\text{sum of remaining})]$
4. Compute intermediate sum: $1.649+2.718+4.482+7.389+12.18+20.09+22.12=70.628$
5. Finalize Trapezoidal: $0.25\times [1+2(70.628)+45.60]=46.964$
6. Apply Simpson's Rule: Calculate $\frac{h}{3}\times [\text{first}+\text{last}+4\times (\text{odd indices})+2\times (\text{even indices})]$
7. Calculate weighted sums: Odd indices sum $=40.431$, Even indices sum $=30.197$
8. Finalize Simpson's: $0.167\times [1+4(40.431)+2(30.197)+45.60]=44.876$