Definite Integrals & Numerical Methods: Trapezoidal Rule, Simpson’s Rule, Error Analysis

TAKEAWAYS:

• Understand the exact vs. approximate evaluation of definite integrals and when numerical methods are necessary.
• Master the trapezoidal rule and Simpson’s 1/3 rule for approximating integrals with given precision.
• Learn the error formulas for these methods and how to estimate them.
• Apply numerical integration to real-world problems like calculating areas under curves, average values, and optimization.
• Compare analytical vs. numerical solutions and their trade-offs in computational efficiency and accuracy.
• Recognize when to use each method based on the function’s behavior and required precision.

---

---

1. Definite Integrals: Exact vs. Approximate Evaluation

Definite integrals represent the signed area under a curve between two limits. While some integrals can be evaluated exactly using antiderivatives, others (especially those involving complex functions) require numerical approximation.

When is Numerical Integration Needed?

• The antiderivative cannot be expressed in elementary functions (e.g., $\int e^{-x^2}dx$).
• The function is given in tabular or experimental form (e.g., sensor data, stock prices).
• High precision is required, and exact methods are computationally expensive.

Example 1: Exact vs. Approximate Evaluation

Consider evaluating ${\int }_0^1\frac{1}{1+x^2}dx$:

• Exact solution: ${\tan }^{-1}(x){|}_0^1=\frac{\pi }{4}\approx 0.7854$.
• Numerical approximation: We will compute this using the trapezoidal rule and compare.

---

2. Numerical Integration Methods

(A) Trapezoidal Rule

The trapezoidal rule approximates the area under a curve by dividing it into trapezoids rather than rectangles (as in Riemann sums).

Formula: For $n$ subintervals of equal width $h=\frac{b-a}{n}$, $${\int }_a^{b}f(x)dx\approx \frac{h}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+\cdots +2f(x_{n-1})+f(x_n)\right],$$ where $x_i=a+ih$.

Worked Example 2: Trapezoidal Rule with 3 Points

Evaluate ${\int }_0^1\frac{1}{1+x^2}dx$ using the composite trapezoidal rule with $n=3$.

Step 1: Divide the interval

• $a=0$, $b=1$, $n=3$ → $h=\frac{1-0}{3}=\frac{1}{3}$.
• Points: $x_0=0$, $x_1=\frac{1}{3}$, $x_2=\frac{2}{3}$, $x_3=1$.

Step 2: Compute function values $$f(x)=\frac{1}{1+x^2}$$ $$f(0)=1,f\left(\frac{1}{3}\right)=\frac{1}{1+{\left(\frac{1}{3}\right)}^2}=\frac{9}{10},f\left(\frac{2}{3}\right)=\frac{9}{13},f(1)=\frac{1}{2}.$$

Step 3: Apply the trapezoidal rule $$\text{Approximation}=\frac{1/3}{2}\left[1+2\left(\frac{9}{10}\right)+2\left(\frac{9}{13}\right)+\frac{1}{2}\right]=\frac{1}{6}\left[1+1.8+1.3846+0.5\right]\approx \mathrm{0.7855.}$$ Exact value: $\frac{\pi }{4}\approx 0.7854$. Error: $|0.7855-0.7854|=0.0001$ (very small for $n=3$).

(B) Simpson’s 1/3 Rule

Simpson’s rule provides a higher accuracy by fitting parabolic arcs to subintervals. It requires an even number of subintervals ($n$ must be even).

Formula: $${\int }_a^{b}f(x)dx\approx \frac{h}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots +2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right],$$ where $h=\frac{b-a}{n}$.

Worked Example 3: Simpson’s Rule with 4 Points

Evaluate the same integral ${\int }_0^1\frac{1}{1+x^2}dx$ using Simpson’s rule with $n=4$.

Step 1: Divide the interval

• $h=\frac{1-0}{4}=\frac{1}{4}$.
• Points: $x_0=0$, $x_1=\frac{1}{4}$, $x_2=\frac{1}{2}$, $x_3=\frac{3}{4}$, $x_4=1$.

Step 2: Compute function values $$f(0)=1,f\left(\frac{1}{4}\right)=\frac{16}{17},f\left(\frac{1}{2}\right)=\frac{4}{5},f\left(\frac{3}{4}\right)=\frac{16}{25},f(1)=\frac{1}{2}.$$

Step 3: Apply Simpson’s rule $$\text{Approximation}=\frac{1/4}{3}\left[1+4\left(\frac{16}{17}\right)+2\left(\frac{4}{5}\right)+4\left(\frac{16}{25}\right)+\frac{1}{2}\right]\approx \mathrm{0.7854.}$$ Exact value: $\frac{\pi }{4}\approx 0.7854$. Error: $|0.7854-0.7854|=0$ (exact for this case due to polynomial nature).

---

3. Error Analysis in Numerical Integration

The error in numerical integration depends on:

1. The method used (trapezoidal vs. Simpson’s).
2. The number of subintervals ($n$).
3. The smoothness of the function (higher derivatives).

Error Formulas

Worked Example 4: Error Estimation for Trapezoidal Rule

For ${\int }_0^1\frac{1}{1+x^2}dx$ with $n=3$:

• $f^{\prime \prime }(x)=\frac{6x^2-2}{(1+x^2{)}^3}$.
• Maximum of $|f^{\prime \prime }(x)|$ on $[0,1]$ occurs at $x=0$: $|f^{\prime \prime }(0)|=2$.
• Error bound: $$|E_T|\le \frac{(1-0{)}^3}{12\cdot 3^2}\cdot 2=\frac{1}{32}\approx \mathrm{0.03125.}$$
• Actual error: $0.0001$ (much smaller than the bound).

---

4. Comparison of Numerical Methods

---

5. Real-World Applications

(A) eSewa and Khalti: Transaction Processing

• Problem: When processing thousands of transactions per second, eSewa and Khalti must approximate the total transaction volume over time.
• Solution: Numerical integration (e.g., trapezoidal rule) is used to estimate total revenue from transaction logs when exact summation is computationally expensive.
• Example: If transaction amounts are recorded at discrete time intervals, the total revenue over an hour can be approximated using: $$\text{Total Revenue}\approx {\int }_0^{3600}f(t)dt\approx \text{Trapezoidal Rule}.$$

(B) Pathao and Ncell: Traffic and Demand Prediction

• Problem: Ride-hailing apps like Pathao need to predict demand in different city zones to optimize driver allocation.
• Solution: Demand data is often sampled at discrete time points. Numerical integration helps estimate the total demand over a time period.
• Example: If demand $D(t)$ is recorded every 15 minutes, the total demand from 6 AM to 10 AM can be approximated as: $${\int }_6^{10}D(t)dt\approx \text{Simpson’s Rule}.$$

(C) NTC and NEPSE: Energy and Stock Price Analysis

• Problem: The Nepal Stock Exchange (NEPSE) and Nepal Telecommunications Corporation (NTC) analyze trends in stock prices and energy consumption.
• Solution: Numerical integration is used to compute:
  1. Total energy consumption over a month from hourly data.
  2. Area under stock price curves to estimate cumulative gains/losses.
• Example: If NEPSE index is recorded daily, the total change over a year can be approximated using: $${\int }_0^{365}P(t)dt\approx \text{Trapezoidal Rule}.$$

(D) Daraz and Amazon: Inventory Optimization

• Problem: E-commerce platforms like Daraz must predict total sales to manage inventory.
• Solution: Sales data (recorded per hour/day) is integrated numerically to forecast monthly revenue.
• Example: If sales $S(t)$ are recorded every 2 hours, the total sales for a day are: $${\int }_0^{24}S(t)dt\approx \text{Simpson’s Rule}.$$

---

6. Worked Example: Real-World Scenario (Bank Loan Interest)

Problem: A bank offers a loan with an interest rate that varies continuously over time. The interest rate at time $t$ (in years) is given by: $$r(t)=0.05+0.01t^2.$$ If the principal amount is Rs. 1,00,000, compute the total interest paid over 5 years using:

1. Exact integration (if possible).
2. Trapezoidal rule with $n=5$.

Solution

1. Exact Integration: $$\text{Total Interest}={\int }_0^{5}100000\cdot r(t)dt=100000{\int }_0^5(0.05+0.01t^2)dt.$$ $$=100000{\left[0.05t+\frac{0.01t^3}{3}\right]}_0^5=100000\left(0.25+\frac{1.25}{3}\right)=100000\times 0.6667=66,667.$$

2. Trapezoidal Rule ($n=5$):

   • $h=\frac{5-0}{5}=1$.
   • Points: $t_0=0$, $t_1=1$, $t_2=2$, $t_3=3$, $t_4=4$, $t_5=5$.
   • $r(t_i)=0.05+0.01t_i^2$: $$r(0)=0.05,r(1)=0.06,r(2)=0.09,r(3)=0.14,r(4)=0.21,r(5)=\mathrm{0.30.}$$
   • Trapezoidal approximation: $$\text{Total Interest}\approx 100000\cdot \frac{1}{2}\left[0.05+2(0.06+0.09+0.14+0.21)+0.30\right]=100000\times 0.665=66,500.$$
   • Error: $|66,667-66,500|=167$ (small for $n=5$).

---

7. Choosing the Right Method

---

8. Common Mistakes to Avoid

1. Forgetting to check if $n$ is even for Simpson’s rule.
2. Incorrectly applying the weights (e.g., using $4$ for the first and last points in Simpson’s rule).
3. Ignoring error bounds—always estimate the error when precision is critical.
4. Miscounting the number of subintervals ($n$ vs. number of points).
5. Assuming Simpson’s rule is always better—it requires more computations and an even $n$.

---

9. Exam Tip

What Examiners Look For

1. Correct Formula Application:

   • Ensure you use the right weights (1, 2, 4 in Simpson’s rule).
   • Do not mix up $n$ (subintervals) with the number of points ($n+1$).

2. Error Analysis:

   • Always state the error formula and compute the error bound.
   • Example: For the trapezoidal rule, write: $$|E_T|\le \frac{(b-a{)}^3}{12n^2}\max |f^{\prime \prime }(x)|.$$

3. Real-World Connection:

   • In exam questions, relate numerical integration to a real scenario (e.g., "This could represent the total revenue for a company over a year").
   • Example answer starter:

     "The trapezoidal rule approximates the total transaction volume for eSewa over a month, where transactions are recorded at discrete time intervals."

4. Comparison Questions:

   • If asked to compare methods, create a table like the one above and highlight accuracy vs. computational cost.

5. Units and Interpretation:

   • Always interpret the result in the context of the problem.
   • Example: If integrating a rate function, the result is a total quantity (e.g., total distance, total revenue).

Model Answer Structure for Exam Questions

Question: Evaluate ${\int }_0^2{e}^{-x^2}dx$ using the trapezoidal rule with $n=4$. Estimate the error.

Model Answer:

1. Step 1: Define $h=\frac{2-0}{4}=0.5$.
2. Step 2: Compute $f(x_i)=e^{-x_i^2}$ for $x_i=0,0.5,1,1.5,2$.
3. Step 3: Apply the trapezoidal formula: $${\int }_0^2{e}^{-x^2}dx\approx \frac{0.5}{2}\left[f(0)+2f(0.5)+2f(1)+2f(1.5)+f(2)\right].$$
4. Step 4: Calculate the numerical value (show computations).
5. Step 5: Estimate the error using: $$|E_T|\le \frac{(2-0{)}^3}{12\cdot 4^2}\max |f^{\prime \prime }(x)|.$$
   • Compute $f^{\prime \prime }(x)=(4x^2-2)e^{-x^2}$ and find its maximum on $[0,2]$.
   • Substitute to get the error bound.

---

10. Summary Table for Quick Revision

---