Numerical Integration: Trapezoidal & Simpson’s Rules

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What is Numerical Integration?

Definite integrals ${\int }_a^{b}f(x)dx$ often have no elementary antiderivative. Numerical integration approximates their value by dividing the area under the curve into simple shapes (rectangles, trapezoids, parabolas) and summing their areas.

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1. Trapezoidal Rule

Definition: Divide $[a,b]$ into $n$ equal subintervals of width $h=\frac{b-a}{n}$. Approximate the area under $f(x)$ as the sum of trapezoids: $$T_n=\frac{h}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+\cdots +2f(x_{n-1})+f(x_n)\right]$$ where $x_i=a+ih$.

How it works:

• Step 1: Choose $n$ (number of strips). Larger $n$ → better accuracy.
• Step 2: Calculate $h=\frac{b-a}{n}$.
• Step 3: Evaluate $f(x)$ at $x_0,x_1,\dots ,x_n$.
• Step 4: Apply the formula.

Worked Example: Approximate ${\int }_0^1{e}^{-x^2}dx$ using $n=4$.

Solution:

1. $h=\frac{1-0}{4}=0.25$.
2. Evaluate $f(x)=e^{-x^2}$ at $x=0,0.25,0.5,0.75,1$:
   • $f(0)=1$
   • $f(0.25)\approx 0.9394$
   • $f(0.5)\approx 0.7788$
   • $f(0.75)\approx 0.5769$
   • $f(1)\approx 0.3679$
3. Apply the formula: $$T_4=\frac{0.25}{2}\left[1+2(0.9394+0.7788+0.5769)+0.3679\right]=0.7468$$ Actual value ≈ 0.7468 (exact to 4 decimal places). Here, $n=4$ gives the exact result by coincidence!

Error Analysis: The error $E_T$ is bounded by: $$|E_T|\le \frac{(b-a)h^2}{12}\underset{a\le x\le b}{\max }|f^{\prime \prime }(x)|$$ For our example, $f^{\prime \prime }(x)=(4x^2-2)e^{-x^2}$, so $|f^{\prime \prime }(x)|\le 2$. Thus: $$|E_T|\le \frac{1\cdot (0.25{)}^2}{12}\cdot 2=0.0083$$ The approximation is accurate to 2 decimal places.

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2. Simpson’s 1/3 Rule

Definition: Requires an even number of subintervals ($n$ even). Approximates the area using parabolic arcs (quadratic functions) over pairs of subintervals: $$S_n=\frac{h}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots +2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right]$$ Pattern: $1,4,2,4,2,\dots ,4,1$.

How it works:

• Step 1: Ensure $n$ is even.
• Step 2: Calculate $h=\frac{b-a}{n}$.
• Step 3: Evaluate $f(x)$ at all $x_i$.
• Step 4: Apply the weighted sum.

Worked Example: Use Simpson’s Rule with $n=4$ to approximate ${\int }_0^1\sqrt{x}dx$.

Solution:

1. $h=\frac{1-0}{4}=0.25$.
2. Evaluate $f(x)=\sqrt{x}$ at $x=0,0.25,0.5,0.75,1$:
   • $f(0)=0$
   • $f(0.25)=0.5$
   • $f(0.5)\approx 0.7071$
   • $f(0.75)\approx 0.8660$
   • $f(1)=1$
3. Apply Simpson’s Rule: $$S_4=\frac{0.25}{3}\left[0+4(0.5)+2(0.7071)+4(0.8660)+1\right]=\frac{0.25}{3}\times 7.3324\approx 0.6110$$ Actual value = $\frac{2}{3}\approx 0.6667$. The error is due to $n=4$ being small.

Error Analysis: The error $E_S$ is bounded by: $$|E_S|\le \frac{(b-a)h^4}{180}\underset{a\le x\le b}{\max }|f^{(4)}(x)|$$ For $f(x)=\sqrt{x}$, $f^{(4)}(x)$ is large near $x=0$, so the error grows if $n$ is too small.

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Comparison: Trapezoidal vs. Simpson’s Rule

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3. Composite Rules

For better accuracy, use more strips:

• Trapezoidal Rule (Composite): $$T_n=\frac{h}{2}\left[f(x_0)+2\sum _{i=1}^{n-1}f(x_i)+f(x_n)\right]$$
• Simpson’s Rule (Composite): $$S_n=\frac{h}{3}\left[f(x_0)+4\sum _{\text{odd }i}f(x_i)+2\sum _{\text{even }i}f(x_i)+f(x_n)\right]$$

Worked Example: Use Simpson’s Rule with $n=6$ to approximate ${\int }_0^1{e}^{x}dx$.

Solution:

1. $h=\frac{1-0}{6}\approx 0.1667$.
2. Evaluate $f(x)=e^x$ at $x=0,0.1667,0.3333,0.5,0.6667,0.8333,1$:
   • $f(0)=1$
   • $f(0.1667)\approx 1.1814$
   • $f(0.3333)\approx 1.3956$
   • $f(0.5)\approx 1.6487$
   • $f(0.6667)\approx 1.9477$
   • $f(0.8333)\approx 2.3012$
   • $f(1)\approx 2.7183$
3. Apply Simpson’s Rule: $$S_6=\frac{0.1667}{3}\left[1+4(1.1814+1.6487+2.3012)+2(1.3956+1.9477)+2.7183\right]$$ $$S_6\approx \frac{0.1667}{3}\times 25.1916\approx 1.3463$$ Actual value = $e-1\approx 1.7183$. Still an error, but closer than Trapezoidal!

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4. Choosing the Right Method

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5. Real-World Applications

Numerical integration is used in:

1. Engineering: Calculating work done by a variable force. $$W={\int }_{x_1}^{x_2}F(x)dx$$
2. Physics: Finding center of mass or probability distributions.
3. Economics: Computing total revenue from a demand curve.
4. Computer Graphics: Rendering smooth curves.

Example in Physics: Find the work done by a spring with force $F(x)=5x$ from $x=0$ to $x=2$ meters.

Solution: $$W={\int }_0^{2}5xdx={\left[\frac{5x^2}{2}\right]}_0^2=10\text{ Joules}$$ But if $F(x)$ is complex (e.g., $F(x)=\sin (x^2)$), use Simpson’s Rule with $n=100$ for an approximation.

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Exam Tip: How to Score Full Marks

1. Show all steps:

   • Write the formula clearly.
   • List all $x_i$ values and $f(x_i)$ calculations.
   • Box your final answer.

2. Label diagrams:

   • Draw the curve and strips (trapezoids/parabolas).
   • Label $a,b,h,x_i$ points.

3. Error analysis (if asked):

   • State the error formula (e.g., $|E_T|\le \frac{(b-a)h^2}{12}\max |f^{\prime \prime }(x)|$).
   • Calculate $\max |f^{\prime \prime }(x)|$ if possible.

4. Common mistakes to avoid:

   • Forgetting to multiply by $\frac{h}{2}$ or $\frac{h}{3}$.
   • Misapplying the pattern (e.g., $4,2,4,2$ vs. $1,4,2,4,1$ in Simpson’s).
   • Using odd $n$ for Simpson’s Rule.

5. NEB-style questions to practice:

   • "Approximate ${\int }_1^2\frac{1}{x}dx$ using Trapezoidal Rule with $n=4$."
   • "Use Simpson’s 1/3 Rule to estimate ${\int }_0^{\pi /2}\sin (x)dx$ with $n=6$."
   • "Compare the errors in Trapezoidal and Simpson’s Rules for ${\int }_0^1{e}^{x}dx$ with $n=4$."

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NEB Board-Style Questions

1. Short Answer:

   • Approximate ${\int }_0^1\ln (1+x)dx$ using the Trapezoidal Rule with $n=4$.
   • Answer: $T_4=\frac{0.25}{2}[0+2(0.2513+0.4055+0.5108)+0.6931]\approx 0.4055$.

2. Long Answer:

   • Use Simpson’s 1/3 Rule with $n=6$ to estimate ${\int }_0^1\sqrt{1+x^2}dx$. Find the percentage error if the exact value is $1.1981$.
   • Answer: $$S_6\approx 1.1826,\text{Error}=\frac{|1.1981-1.1826|}{1.1981}\times 100\%\approx 1.29\%$$

3. Theoretical:

   • Derive the error formula for the Trapezoidal Rule. Why is Simpson’s Rule more accurate?
   • Answer: The Trapezoidal Rule approximates $f(x)$ linearly, leading to $O(h^2)$ error. Simpson’s Rule uses quadratics, reducing error to $O(h^4)$ because it fits more points.

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Final Note: Numerical integration is about approximation, not exactness. Always check:

• Is $n$ large enough?
• Does the method match the function’s smoothness?
• Are the calculations precise?

Practice with both rules and compare results!