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.