Numerical integration (integral, trapz) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the integral function in MATLAB do?

The integral function calculates the numerical integral (area under the curve) of a function over a specified interval using adaptive quadrature methods.

beginner

How does the trapz function approximate an integral?

trapz uses the trapezoidal rule, which approximates the area under a curve by dividing it into trapezoids and summing their areas.

beginner

Syntax of integral function in MATLAB?

Basic syntax: q = integral(fun, a, b) where fun is the function handle, and a, b are the integration limits.

intermediate

When should you prefer trapz over integral?

Use trapz when you have discrete data points and want to approximate the integral, especially if the function is not easily expressed as a function handle.

intermediate

What is the main difference between integral and trapz in MATLAB?

integral works with function handles and uses adaptive methods for higher accuracy, while trapz works with vectors of data points and uses the trapezoidal rule.

Which MATLAB function uses adaptive quadrature to compute integrals?

A diff

B integral

C sum

D trapz

integral uses adaptive quadrature, while trapz uses the trapezoidal rule.

What input does trapz require to approximate an integral?

A Matrix multiplication

B Function handle

C Symbolic expression

D Discrete data points

trapz approximates integrals from discrete data points.

What does the trapezoidal rule approximate the area under a curve with?

A Trapezoids

B Circles

C Rectangles

D Triangles

The trapezoidal rule approximates the area using trapezoids.

Which function is better for integrating a smooth function defined by a formula?

A plot

B trapz

C integral

D diff

integral is better for smooth functions defined by formulas.

What is the correct syntax to integrate function f from 0 to 1 using integral?

A integral(f, 0, 1)

B trapz(f, 0, 1)

C integral(0, 1, f)

D trapz(0, 1, f)

The syntax is integral(f, 0, 1) where f is a function handle.

Explain how the integral and trapz functions differ in MATLAB and when to use each.

Think about the input types and accuracy methods.

Describe the trapezoidal rule and how trapz applies it to approximate integrals.

Imagine connecting points with straight lines and calculating the area.

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