# Integration and Differentiation in MATLAB: Video Tutorial

URL: https://yarpiz.com/integration-and-differentiation-video-tutorial-yptnm190520s12
Author: Mostapha Kalami Heris
Published: 2019-07-03
Updated: 2026-10-30
Topics: Courses

> Compute integrals and derivatives in MATLAB using functions or sampled data. Explore quadrature, trapezoidal integration, gradients, tolerances and sample spacing.

This video tutorial reviews and implements integration and differentiation in MATLAB. It is part of the Numerical Computations course. To watch the full course, [visit the course page](https://yarpiz.com/numerical-computations-in-matlab-video-tutorial-572).

## Two kinds of problems

The four videos cover two situations:

- **You have a function.** The integral $\int_a^b f(x)\,dx$ is computed by adaptive quadrature: the solver evaluates $f$ at chosen points and refines the intervals until the error is small enough. MATLAB offers `integral` and `quadgk` for one variable, `integral2` and `quad2d` for double integrals, and `integral3` for triple integrals.
- **You have samples.** When only values $y_i$ at points $x_i$ are known, sums replace integrals and differences replace derivatives. MATLAB offers `sum`, `cumsum`, `trapz` and `cumtrapz` for integration, and `diff`, `gradient` and `del2` for differentiation.

## Watch online

The four sections of this tutorial are available on YouTube and are embedded here as a playlist.

::: video {playlist="PLgH3sgdvgO4TnJLwU0NIwcr2Xd8iMQTL5" youtube="wiYeBfBBEys" count="4" title="Integration and Differentiation in MATLAB: Video Tutorial"}
:::

## What each video covers

### Section 1: Numerical integration using `integral` and `quadgk`

Watch this part on [YouTube](https://youtu.be/wiYeBfBBEys).

The basic call is `integral(f, a, b)`, where `f` is a function handle. The video starts with $\int_1^2 x\,dx = 1.5$, then integrates $x e^{-x^2}$ from 0 to 1 (about $0.3161$) and from 0 to infinity (exactly $1/2$). The integrand uses element-wise operators (`.*`, `./`, `.^`), because `integral` evaluates it on many points at once.

```matlab {title="Integrate over an infinite interval"}
f = @(x) x.*exp(-x.^2);
q = integral(f, 0, Inf)
```

Next comes $\int_0^1 \ln x\,dx = -1$, which has a singularity at the lower limit. The video uses this example to explain the error tolerances. If $Q$ is the exact value and $q$ the computed one, the solver tries to satisfy

$$
|Q - q| \le \max\left(\text{AbsTol},\ \text{RelTol} \cdot |q|\right)
$$

Setting `'RelTol'` to 0 leaves only the absolute tolerance in control. A smaller `'AbsTol'` requests a tighter error estimate, but it does not remove floating point limits or guarantee that every difficult integrand reaches that accuracy. Check any warning from the solver.

The video then computes contour integrals in the complex plane. The integral of $1/z$ around the unit circle is $2\pi i$ (Cauchy's theorem says it would be zero if the path enclosed no pole). It is first computed with the parametrization $\gamma(t) = \cos t + i \sin t$:

$$
\oint_C f(z)\,dz = \int_{0}^{2\pi} f\big(\gamma(t)\big)\,\gamma'(t)\,dt
$$

A simpler way uses the `'Waypoints'` option: the path starts and ends at 1 and passes through $i$, $-1$ and $-i$. A triangular path through $2 + i$ and $2 - i$ does not enclose the pole at the origin, and the result is almost zero.

Finally, `quadgk` (adaptive Gauss-Kronrod quadrature) is called in the same way and gives the same results. It accepts the same tolerance and waypoint options, plus `'MaxIntervalCount'`.

### Section 2: Double and triple integrals

Watch this part on [YouTube](https://youtu.be/e1zyt26NBZ4).

`integral2(f, xmin, xmax, ymin, ymax)` and `quad2d` compute double integrals, and `integral3` computes triple integrals. The limits of $x$ are numbers, while the limits of $y$ (and $z$) can be numbers or function handles of the outer variables. The video solves three examples:

$$
\int_{-\pi}^{2\pi}\!\int_{0}^{\pi} \left(y \sin x + x \cos y\right) dy\,dx = -\pi^2
$$

$$
\int_{0}^{1}\!\int_{0}^{1-x} \frac{dy\,dx}{\sqrt{x+y}\,\left(1+x+y\right)^2} = \frac{\pi}{4} - \frac{1}{2}
$$

$$
\int_{0}^{\pi}\!\int_{0}^{1}\!\int_{-1}^{1} \left(y \sin x + z \cos x\right) dz\,dy\,dx = 2
$$

In the second integral, the upper limit of $y$ depends on $x$, so it is passed as a function handle:

```matlab {title="Integrate over a triangular region"}
g = @(x, y) 1./(sqrt(x + y).*(1 + x + y).^2);
ymax = @(x) 1 - x;
q = integral2(g, 0, 1, 0, ymax)
```

### Section 3: Discrete integrals and summations

Watch this part on [YouTube](https://youtu.be/Fak-n-SETpU).

In discrete systems, a sum plays the role of an integral. The video shows `sum` and `cumsum` on vectors and on a matrix along either dimension, and `sum(Y(:))` for the total of all elements.

It then approximates $\int_0^{\pi/2} \sin x\,dx = 1$ with 100 samples. The plain sum of the samples times $\Delta x$ is close to 1. `trapz` uses the trapezoidal rule and gives a better result. With equal spacing you can pass the spacing $\Delta x$ or multiply by it. With unequal spacing you must pass the vector of $x$ values. `cumtrapz` returns the running integral, which the video plots against $1 - \cos x$. Nested calls of `trapz` give double and triple integrals.

```matlab {title="Integrate sampled sine values"}
x = linspace(0, pi/2, 100);
y = sin(x);
q = trapz(x, y)
```

### Section 4: Numerical differentiation

Watch this part on [YouTube](https://youtu.be/g93udhjFFFw).

`diff` is the opposite of `cumsum`: for a vector it returns the differences $x_{k+1} - x_k$, so the result has one element less. The derivative of $\sin x$ on 100 points from 0 to $2\pi$ is then approximated by the forward difference

$$
y'(x_k) \approx \frac{y_{k+1} - y_k}{x_{k+1} - x_k}
$$

and compared with the exact derivative, $\cos x$. Reducing the spacing improves the approximation in this smooth example until rounding error becomes important. Differencing noisy measurements can amplify the noise.

`gradient` returns a result of the same length as the input and gives a smaller error than `diff` in this example. It also works in several dimensions. For $f(x, y) = x\,e^{-(x^2 + y^2)}$ on a grid with spacing $h$, `[gx, gy] = gradient(Z, h)` returns both partial derivatives, and `quiver` draws the gradient vectors over a `contour` plot.

The last function is `del2`, which is related to the Laplacian

$$
\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}
$$

`del2` returns $\nabla^2 f / (2N)$, where $N$ is the number of dimensions. For a function of two variables, the video multiplies the result by 4 to get the Laplacian.

## Applications

- Areas, volumes and other quantities defined by integrals that have no closed form.
- Integrating measured data, such as velocity samples, to get a total or a running total.
- Estimating rates of change and gradients from sampled or grid data.

## Check units and spacing

Pass the actual sample coordinates when measurements are not evenly spaced. A raw sum or difference does not include the sample spacing, so it does not by itself have the units of an integral or derivative. For a new integrand, compare tolerances and check against an analytic result when one is available. For sampled data, repeat the calculation on a finer grid when possible and check whether the result stabilizes.

## How to cite

Mostapha Kalami Heris, Integration and Differentiation in MATLAB: Video Tutorial (YPTNM190520S12), Yarpiz, 2019. https://yarpiz.com/integration-and-differentiation-video-tutorial-yptnm190520s12