Numerical integration (integral, trapz) in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of trapezoidal integration with uneven spacing

What is the output of the following MATLAB code snippet?

MATLAB

x = [0 1 2 4];
y = [0 1 4 16];
result = trapz(x, y);
disp(result);

A 23

B 18

C 20

D 22

💡 Hint

Recall that trapz integrates using trapezoids between points with given x spacing.

The trapezoidal rule sums areas of trapezoids between points. Here, intervals are [0,1], [1,2], [2,4]. Calculating each area and summing gives 23.

❓ Predict Output

intermediate

Integral function output for sine over 0 to pi

What is the output of this MATLAB code?

MATLAB

f = @(x) sin(x);
result = integral(f, 0, pi);
disp(result);

A 2

B 1

C 0

D 3.1416

💡 Hint

The integral of sin(x) from 0 to pi is a known exact value.

The integral of sin(x) from 0 to pi is 2, which is the exact area under the sine curve in that interval.

🔧 Debug

advanced

Identify the error in trapezoidal integration code

What error does this MATLAB code produce when run?

MATLAB

x = 0:0.1:1;
y = x.^2;
result = trapz(y, x);
disp(result);

A Error: trapz expects x as first argument and y as second.

B The code runs and outputs 0.6650

C Error: Dimensions of arrays being concatenated are not consistent.

D Error: Inputs must be vectors of the same length.

💡 Hint

Check the order and size of inputs to trapz function.

trapz(X, Y) computes the integral of Y with respect to X. Here, the order is reversed causing it to compute integral of x with respect to y (0.6650) instead of y with respect to x (approx. 0.3350).

🧠 Conceptual

advanced

Comparing integral and trapz accuracy

Which statement best describes the difference between MATLAB's integral and trapz functions?

A trapz uses adaptive quadrature; integral uses fixed trapezoidal sums.

B Both integral and trapz use the same algorithm but integral is faster.

C integral uses adaptive quadrature for higher accuracy; trapz uses simple trapezoidal sums which may be less accurate.

D integral only works for symbolic functions; trapz only works for numeric vectors.

💡 Hint

Consider how each function approaches numerical integration.

integral uses adaptive quadrature methods that adjust evaluation points for accuracy, while trapz uses fixed trapezoidal sums over given points, which can be less accurate for complex functions.

🚀 Application

expert

Calculate integral of exp(-x^2) from -1 to 1 using trapz

Given the function f(x) = exp(-x^2), which option correctly computes the integral from -1 to 1 using trapz with 1000 points?

A

x = -1:0.002:1;
y = exp(-x.^2);
result = trapz(y,x);

B

x = linspace(-1,1,1000);
y = exp(-x.^2);
result = trapz(y,x);

C

x = linspace(-1,1,1000);
y = exp(-x^2);
result = trapz(x,y);

D

x = linspace(-1,1,1000);
y = exp(-x.^2);
result = trapz(x,y);

💡 Hint

Remember trapz syntax: trapz(x, y) where x and y are vectors of same length.

Option D correctly defines x and y vectors with element-wise power and uses trapz(x,y). Option D swaps arguments causing error. Option D uses ^ instead of .^ causing error. Option D swaps arguments and uses colon operator with step size 0.002 but swaps arguments causing error.

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