Numerical integration (integral, trapz) in MATLAB - Mini Project: Build & Apply

Numerical integration using integral and trapz in MATLAB

📖 Scenario: You are working as a data analyst and need to calculate the area under a curve representing a physical measurement over time. This is a common task in science and engineering to find total quantities from rate data.

🎯 Goal: Build a MATLAB script that calculates the integral of a function using both the integral function and the trapz function for numerical integration. You will compare the results to understand how these methods work.

📋 What You'll Learn

Create a vector of x values from 0 to 10 with 100 points

Create a vector y with the function values y = sin(x)

Use the integral function to calculate the integral of sin(x) from 0 to 10

Use the trapz function to calculate the integral approximation from the x and y vectors

Print both results to compare

💡 Why This Matters

🌍 Real World

Numerical integration is used in physics, engineering, and data analysis to find total quantities like distance from speed or area under curves from measurements.

💼 Career

Understanding numerical integration helps in roles like data analyst, engineer, or scientist where you work with experimental data or simulations.

1

Create the x vector

Create a vector called x with 100 points linearly spaced from 0 to 10 using the linspace function.

MATLAB

% Create the x vector with 100 points from 0 to 10
% Your code here

Hint

Use linspace(0, 10, 100) to create 100 points between 0 and 10.

2

Calculate y = sin(x)

Create a vector called y that contains the sine of each value in x using the sin function.

MATLAB

x = linspace(0, 10, 100);
% Calculate y as sin(x)
% Your code here

Hint

Use y = sin(x); to get sine values for each x.

3

Calculate integrals using integral and trapz

Calculate the integral of sin(x) from 0 to 10 using the integral function and store it in int_val. Then calculate the integral approximation using trapz with x and y and store it in trapz_val.

MATLAB

x = linspace(0, 10, 100);
y = sin(x);
% Calculate int_val using integral
% Calculate trapz_val using trapz
% Your code here

Hint

Use integral(@sin, 0, 10) and trapz(x, y) to calculate integrals.

4

Display the results

Print the values of int_val and trapz_val using fprintf to show the integral results with 6 decimal places.

MATLAB

x = linspace(0, 10, 100);
y = sin(x);
int_val = integral(@sin, 0, 10);
trapz_val = trapz(x, y);
% Print int_val and trapz_val
% Your code here

Hint

Use fprintf('Integral using integral: %.6f\n', int_val); and similarly for trapz_val.

Practice

1. What does the MATLAB function integral primarily do?

easy

A. Finds the derivative of a function

B. Calculates the numerical integral of a function over a specified interval

C. Plots data points on a graph

D. Solves algebraic equations symbolically

Solution

1. Step 1: Understand the purpose of integral

   The integral function in MATLAB is used to calculate the area under a curve defined by a function between two limits.

2. Step 2: Compare with other options

   Options A, B, and C describe different MATLAB functions or operations unrelated to numerical integration.

3. Final Answer:

   Calculates the numerical integral of a function over a specified interval -> Option B

4. Quick Check:

   integral = numerical integration [OK]

Hint: Remember integral finds area under curve from function [OK]

Common Mistakes:

• Confusing integral with symbolic math functions
• Thinking integral plots graphs
• Mixing integral with derivative functions

2. Which of the following is the correct syntax to compute the integral of sin(x) from 0 to pi using integral in MATLAB?

easy

A. integral(@sin, 0, pi)

B. integral(sin(x), 0, pi)

C. integral('sin(x)', 0, pi)

D. integral(sin, [0 pi])

Solution

1. Step 1: Identify correct function handle syntax

   In MATLAB, integral requires a function handle, written as @functionName. So @sin is correct.

2. Step 2: Check argument format

   The limits are given as separate arguments, so 0, pi is correct. Other options misuse syntax by passing strings or symbolic expressions.

3. Final Answer:

   integral(@sin, 0, pi) -> Option A

4. Quick Check:

   Use @ for function handle in integral [OK]

Hint: Use @ before function name for integral [OK]

Common Mistakes:

• Passing function as string instead of handle
• Using symbolic expressions instead of function handle
• Putting limits inside brackets as one argument

3. What is the output of the following MATLAB code?

x = 0:0.5:2;
y = x.^2;
area = trapz(x, y);
disp(area);

medium

A. 2.5

B. 4.0

C. 3.0

D. 2.75

Solution

1. Step 1: Calculate y values

   Given x = [0 0.5 1 1.5 2], y = x.^2 = [0 0.25 1 2.25 4].

2. Step 2: Apply trapezoidal rule with trapz

   trapz approximates integral as sum of trapezoids:
   Area ≈ 0.5*((0+0.25)/2 + (0.25+1)/2 + (1+2.25)/2 + (2.25+4)/2)
   = 0.5*(0.125 + 0.625 + 1.625 + 3.125) = 0.5*5.5 = 2.75

3. Final Answer:

   2.75 -> Option D

4. Quick Check:

   trapz(x,y) ≈ 2.75 [OK]

Hint: Calculate y, then sum trapezoids with trapz [OK]

Common Mistakes:

• Forgetting to square x for y
• Using incorrect step size in calculation
• Mixing up trapz and integral outputs

4. Identify the error in this MATLAB code snippet:

x = linspace(0,1,5);
y = x.^3;
result = trapz(y, x);

medium

A. Using .^3 is incorrect for element-wise power

B. linspace should have 6 points instead of 5

C. Arguments to trapz are reversed; should be trapz(x, y)

D. trapz cannot be used with vectors

Solution

1. Step 1: Check trapz argument order

   trapz expects x values first, then y values. Here, arguments are reversed.

2. Step 2: Validate other parts

   linspace with 5 points is valid; .^3 is correct for element-wise cube; trapz works with vectors.

3. Final Answer:

   Arguments to trapz are reversed; should be trapz(x, y) -> Option C

4. Quick Check:

   trapz(x,y) correct argument order [OK]

Hint: Remember trapz(x, y), x first then y [OK]

Common Mistakes:

• Swapping x and y in trapz
• Thinking linspace points count causes error
• Misunderstanding element-wise power operator

5. You have noisy data points x = 0:0.1:1 and y = sin(2*pi*x) + 0.1*randn(size(x)). Which method is best to estimate the integral of the underlying sine wave over [0,1]?

hard

A. Use trapz(x, y) to approximate integral from noisy data

B. Use trapz(y, x) ignoring x order

C. Use integral(@(t) sin(2*pi*t), 0, 1) for exact integral

D. Use symbolic integration on noisy data

Solution

1. Step 1: Understand data and goal

   The data y includes noise, so exact function integral is not possible from data alone.

2. Step 2: Choose appropriate method

   Using trapz(x, y) approximates the integral from discrete noisy data points effectively.

3. Step 3: Evaluate other options

   Use integral(@(t) sin(2*pi*t), 0, 1) for exact integral uses exact function without noise, not data-based; Use trapz(y, x) ignoring x order reverses arguments; Use symbolic integration on noisy data is invalid for numeric noisy data.

4. Final Answer:

   Use trapz(x, y) to approximate integral from noisy data -> Option A

5. Quick Check:

   trapz(x,y) best for noisy data integration [OK]

Hint: Use trapz with noisy data, integral needs exact function [OK]

Common Mistakes:

• Using integral on noisy data without function
• Swapping arguments in trapz
• Trying symbolic integration on numeric noise