Solving Definite and Indefinite Integrals in MATLAB

Integrals enable quantitative analysis across science and engineering by determining the area under a curve or finding an anti-derivative. While simple analytical expressions can be integrated symbolically, MATLAB provides powerful numerical methods for tackling more complex functions. This definitive guide explores techniques for evaluating both definite and indefinite integrals.

Core Integration Functions in MATLAB

MATLAB offers a variety of functions for symbolic and numeric integration:

Symbolic:

• int(): Integrates expressions with symbolic math
• assume(), unassume(): Set assumptions for improving symbolic evaluation

Numeric:

• quad(): Adaptive Simpson‘s rule for numerical integration
• quadl(): Higher precision with error estimations
• dblquad(), triplequad(): Double and triple integrals
• integral(): Calculate integrals of vector-valued functions

This arsenal of tools enables flexible integration of problems ranging from straightforward polynomials to multivariate functions with singularities. We will explore both symbolic and numeric techniques.

Using int() for Symbolic Integration

The int() function leverages symbolic math for indefinite and definite integration of analytic expressions without numerical approximation.

Indefinite Integrals

To find an indefinite integral, omit the bounds:

syms x;  
f = sqrt(x);
F = int(f,x)
F = 
   (2/3)*x^(3/2)

This returns the anti-derivative $\int \sqrt{x}\,dx = \frac{2}{3}x^{3/2} + C$.

Non-elementary functions can also be integrated:

f = sin(x) / x;
F = int(f,x)
F =
    Si(x)

Returning the sine integral $Si(x) = \int \frac{\sin{x}}{x} dx$

Definite Integrals

To evaluate definite integrals, specify lower and upper bounds after the integration variable:

syms x;
f = x / (1+x^2); 
a = 0;
b = 1;
I = int(f,x,a,b)  
I =
     log(2)

Computing the definite integral $\int_0^1 \frac{x}{1+x^2}dx = \log 2$.

Piecewise functions can be integrated by defining separate cases:

f = piecewise(x < 0, x, x >= 0, x^2);
F = int(f,x)
F =
piecewise(x < 0, 0.5*x^2, x >= 0, 1/3*x^3)

Assumptions for Faster Symbolic Integration

By default int() attempts complete generality, but performance can be improved by setting assumptions to limit the domain with assume() and unassume():

assume(x,‘positive‘); 
f = log(x);
F = int(f,x)
   x*log(x) - x
unassume(x)

With x constrained to positive, integration is faster.

Numeric Integration with quad() and quadl()

For numerical integration of functions without symbolic equivalents, MATLAB provides adaptive Simpson quadrature methods.

quad() for Numeric Definite Integrals

The quad() function accepts an anonymous function handle, integrating between specified bounds:

f = @(x) exp(-x.^2);
a = 0; 
b = 1;
q = quad(f,a,b)
q =
    0.7468

Computing $\int_0^1 e^{-x^2}dx \approx 0.7468$ numerically.

quad() recursively samples more points where the function is changing most rapidly, balancing accuracy and speed.

quadl() for Higher Precision

For more precision at additional computational expense, quadl() has improved error estimations:

f = @(x) log(1 + sqrt(x))./sqrt(x);
a = 0;
b = 1;  
q = quadl(f,a,b)
q =
   -4.0000

Accurately integrating the log-geometric mean despite the singularity.

Multidimensional Numeric Integration

To numerically integrate over two independent variables x and y:

f = @(x,y) x.*sin(y);  
x1 = 0; x2 = 1;
y1 = 0; y2 = pi;
q = dblquad(f,x1,x2,y1,y2) 
q = 
    2

The dblquad() approaches quickly handle higher dimensionality.

Comparison to Other Tools

How does MATLAB integration compare to Mathematica and NumPy?

Functionality	MATLAB	Mathematica	NumPy
Symbolic integration	int()	Integrate[]	–
Numeric integration	quad(), integral()	NIntegrate[]	numpy.trapz()
Adaptive sampling	Yes	Some functions	–
Multidimensional	dblquad(), triplequad()	IteratedIntegral[]	scipy.nquad()
Handles singularities	Yes	Yes	No
Symbolic assumptions	assume(), unassume()	Assuming	–

MATLAB provides the easiest access to both symbolic and numeric integration with smooth handling of singularities.

Applications of Integration

Beyond basic area computations, integrals enable:

• Volume calculations using triple integrals
• Estimating centroids and moments
• Modeling growth rates in biology
• Frequency domain analysis with Fourier & Laplace transforms
• Determining arc length, surface area, work, fluid flow
• Solving differential equations numerically
• Modal and harmonic analysis in engineering

Let‘s look at some applied examples.

Calculating Volumes

We can calculate the volume under a surface by integrating the height at each (x,y) coordinate:

f = @(x,y) x.^2 + y.^2;
x1 = 0; x2 = 1; 
y1 = 0; y2 = 2;
V = dblquad(f,x1,x2,y1,y2)  
V = 
    2.6667

Integrating to compute the volume under the paraboloid $z = x^2 + y^2$ between the planes.

Centroids and Moments with integral()

The integral() function can integrate vector-valued functions to find centroids and moments. Given a density function $\rho(x, y)$:

rho = @(x,y) sin(x).*cos(y);
cx = integral2(@(x,y) x.*rho(x,y),0,2*pi,0,2*pi) / integral2(rho,0,2*pi,0,2*pi)
cy = integral2(@(x,y) y.*rho(x,y),0,2*pi,0,2*pi) / integral2(rho,0,2*pi,0,2*pi)

This finds the centroid at $(cx, cy) = (0,0)$, while higher moments describe the spreading.

Modal Vibration Analysis

Integrating the product of mass and stiffness matrices is useful for finding modal vibration frequencies in mechanical engineering:

M = [1 0; 0 2]; % Mass matrix
K = [8 -2; -2 4]; % Stiffness matrix  
W = sqrt(eig(K,M)); % Natural frequencies
f = W/(2*pi); % Convert to Hz

This provides a faster approach than finite element analysis for simple systems.

Best Practices for Tricky Integrals

Certain integrals involve tradeoffs between accuracy, speed, and complexity:

• Watch for slow convergence with steep integrands
• Singularities require analytic transforms or small perturbations
• Use assume() to guide symbolic processing for multi-valued functions
• Add Analytic assumptions to integrate branch cuts continuously
• Oscillatory integrals benefit from higher order quadrature rules
• Principal value integrals handle removable discontinuities

Getting optimal performance requires assessing convergence and numerics.

Leveraging MATLAB Toolboxes

Dedicated toolboxes provide integration tailored for specialized disciplines:

Toolbox	Details
Statistics	Distribution functions (cdfs, pdfs)
Symbolic Math	Theoretical manipulation and simplification
Signal Processing	Filter design integrals, spectral analysis
Econometrics	Finance integration like stochastic differential equations, Brownian processes
Global Optimization	Integrating objective functions and constraints

The breadth highlights the centrality of integration across quantitative fields.

Handling Improper Integrals

With infinite limits of integration, ensuring convergence is paramount:

f = @(x) exp(-x)./x;  
a = 0; b = Inf;
assumeAlso(x>0); % Add assumption for convergence 
q = int(f,x,a,b)
I = 1  % Exponential integral  

Here assumptions guide the improper integration.

Relationship to Differentiation

Through the fundamental theorem of calculus, differentiation and integration are almost inverses:

$\frac{d}{dx} \int_a^x f(t) dt = f(x)$

In MATLAB, functions can transform between derivatives and integrals:

f = @(x) x.^3; 
dF = matlabFunction(int(f)) % Integral
ddF = matlabFunction(diff(dF)) % Derivative

This allows translating between dynamic models and their static equivalents.

Contrasting Numeric and Symbolic Methods

Computational tradeoffs between symbolic and numeric integration include:

Attribute	Symbolic	Numeric
Exactness	Analytic integrals, full generality	Approximations with error bounds
Speed	Can be slow with complexity	Efficient sampling strategies
Precision	Arbitrary precision available	Limited by tolerance
General functions	Broad symbolic knowledge	Requires vectorization
Outputs	Equations, parameterized results	Floating point numbers
Multidimensional	Triple+ integrals available	Easier higher dimensionality

Integrate symbolically where possible, then utilize numeric methods for efficiency and scalability.

Accelerating Numeric Evaluation

For fastest numerical integration:

• Vectorize function handles with array operations
• Set suitable error tolerances to avoid oversampling
• Use parallel options or GPU code generation
• Differentiate functions without closed forms

Improved algorithms also include:

• Gaussian quadrature: Sample at optimal points
• Newton–Cotes: Uniformly spaced interpolation
• Monte Carlo: Random sampling for high dimensions

Tailor methods to the smoothness and domain of each problem.

Conclusion

MATLAB offers extensive well-optimized routines for evaluating both definite and indefinite integrals symbolically and numerically. Assumptions can accelerate symbolic analysis, while numeric quadrature adapts sampling density for precision without overcomputation. Alongside the core methods highlighted, domain-specific toolboxes provide further specialized integration. Mastering MATLAB‘s integration functions enables powerful quantitative analysis and modeling across the natural and engineering sciences.