Numerical integration (integral, trapz) in MATLAB - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to calculate the integral of sin(x) from 0 to pi using the integral function.

MATLAB

result = integral(@(x) sin(x), 0, [1]);

Drag options to blanks, or click blank then click option'

Api

B2*pi

C0

Dpi/2

Attempts:

3 left

💡 Hint

Common Mistakes

Using 2*pi as the upper limit instead of pi.

Setting the upper limit to 0 which results in zero area.

The integral of sin(x) from 0 to pi is calculated by setting the upper limit to pi.

2fill in blank

medium

Complete the code to approximate the integral of cos(x) from 0 to pi using the trapz function with 100 points.

MATLAB

x = linspace(0, pi, 100);
y = cos(x);
result = trapz(x, [1]);

Drag options to blanks, or click blank then click option'

Acos(x).^2

Bx

Cy

Dsin(x)

Attempts:

3 left

💡 Hint

Common Mistakes

Passing x instead of y to trapz.

Passing an unrelated function like sin(x).

The trapz function integrates the values in y over x. So we pass y as the second argument.

3fill in blank

hard

Fix the error in the code to correctly compute the integral of exp(-x^2) from -1 to 1 using integral.

MATLAB

result = integral(@(x) exp(-[1].^2), -1, 1);

Drag options to blanks, or click blank then click option'

AX

Bt

Cy

Dx

Attempts:

3 left

💡 Hint

Common Mistakes

Using uppercase X instead of lowercase x.

Using a different variable name like t or y.

The anonymous function uses x as the input variable, so the variable inside the function must be x.

4fill in blank

hard

Fill both blanks to create a vector of squares for x from 1 to 5, including only values greater than 4, using logical indexing. This can be useful for selecting weights or points in numerical integration.

MATLAB

x = 1:5;
values = x(x [2] 4).[1]2;

Drag options to blanks, or click blank then click option'

A.^

B>

C<

D*

Attempts:

3 left

💡 Hint

Common Mistakes

Using * instead of .^ for element-wise power.

Using < instead of > in the condition.

Use .^ for element-wise power and > to filter values greater than 4.

5fill in blank

hard

Fill all three blanks to create a structure with fields as uppercase letters of x, values as y, and filter y values greater than 0.5. This can store selected significant function values for numerical integration points.

MATLAB

result = struct();
for i = 1:length(x)
    if y(i) [1] 0.5
        field = upper([2](i));
        result.(field) = [3](i);
    end
end

Drag options to blanks, or click blank then click option'

A>

Bx

Cy

D<

Attempts:

3 left

💡 Hint

Common Mistakes

Using < instead of > in the condition.

Using y instead of x for the field name.

Assigning x(i) instead of y(i) as the value.

The condition checks if y(i) is greater than 0.5, the field name is uppercase of x(i), and the value assigned is y(i).

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