Integration

This example shows how to compute definite integrals using Symbolic Math Toolbox™.

Definite Integral

Show that the definite integral $\int_{a}^{b} f (x) d x$ for $f (x) = s i n (x)$ on $[\frac{π}{2}, \frac{3 π}{2}]$ is 0.

syms x
int(sin(x),pi/2,3*pi/2)

ans =  $0$

Definite Integrals in Maxima and Minima

To maximize $F (a) = \int_{- a}^{a} \sin (a x) \sin (x / a) d x$ for $a \geq 0$ , first, define the symbolic variables and assume that $a \geq 0$ :

syms a x
assume(a >= 0);

Then, define the function to maximize:

F = int(sin(a*x)*sin(x/a),x,-a,a)

F = 
 ${\begin{array}{cl} 1 - \frac{\sin (2)}{2} & if a = 1 \\ \frac{2 a (\sin (a^{2}) \cos (1) - a^{2} \cos (a^{2}) \sin (1))}{a^{4} - 1} & if a \neq 1 \end{array}$

Note the special case here for $a = 1$ . To make computations easier, use assumeAlso to ignore this possibility (and later check that $a = 1$ is not the maximum):

assumeAlso(a ~= 1);
F = int(sin(a*x)*sin(x/a),x,-a,a)

F = 
 $\frac{2 a (\sin (a^{2}) \cos (1) - a^{2} \cos (a^{2}) \sin (1))}{a^{4} - 1}$

Create a plot of $F$ to check its shape:

fplot(F,[0 10])

Figure contains an axes object. The axes object contains an object of type functionline.

Use diff to find the derivative of $F$ with respect to $a$ :

Fa = diff(F,a)

Fa = 
 $\begin{array}{l} \frac{2 σ_{1}}{a^{4} - 1} + \frac{2 a (2 a \cos (a^{2}) \cos (1) - 2 a \cos (a^{2}) \sin (1) + 2 a^{3} \sin (a^{2}) \sin (1))}{a^{4} - 1} - \frac{8 a^{4} σ_{1}}{{(a^{4} - 1)}^{2}} \\ where \\ σ_{1} = \sin (a^{2}) \cos (1) - a^{2} \cos (a^{2}) \sin (1) \end{array}$

The zeros of $F a$ are the local extrema of $F$ :

hold on
fplot(Fa,[0 10])
grid on

Figure contains an axes object. The axes object contains 2 objects of type functionline.

The maximum is between 1 and 2. Use vpasolve to find an approximation of the zero of $F a$ in this interval:

a_max = vpasolve(Fa,a,[1,2])

a_max =  $1.5782881585233198075558845180583$

Use subs to get the maximal value of the integral:

F_max = subs(F,a,a_max)

F_max =  $0.36730152527504169588661811770092 \cos (1) + 1.2020566879911789986062956284113 \sin (1)$

The result still contains exact numbers $\sin (1)$ and $\cos (1)$ . Use vpa to replace these by numerical approximations:

vpa(F_max)

ans =  $1.2099496860938456039155811226054$

Check that the excluded case $a = 1$ does not result in a larger value:

vpa(int(sin(x)*sin(x),x,-1,1))

ans =  $0.54535128658715915230199006704413$

Multiple Integration

Numerical integration over higher dimensional areas has special functions:

integral2(@(x,y) x.^2-y.^2,0,1,0,1)

ans = 
4.0127e-19

There are no such special functions for higher-dimensional symbolic integration. Use nested one-dimensional integrals instead:

syms x y
int(int(x^2-y^2,y,0,1),x,0,1)

ans =  $0$

Line Integrals

Define a vector field F in 3D space:

syms x y z
F(x,y,z) = [x^2*y*z, x*y, 2*y*z];

Next, define a curve:

syms t
ux(t) = sin(t);
uy(t) = t^2-t;
uz(t) = t;

The line integral of F along the curve u is defined as $\int f \cdot d u = \int f (u x (t), u y (t), u z (t)) \cdot \frac{d u}{d t} d t$ , where the $\cdot$ on the right-hand-side denotes a scalar product.

Use this definition to compute the line integral for $t$ from $[0, 1]$

F_int = int(F(ux,uy,uz)*diff([ux;uy;uz],t),t,0,1)

F_int = 
 $\frac{19 \cos (1)}{4} - \frac{\cos (3)}{108} - 12 \sin (1) + \frac{\sin (3)}{27} + \frac{395}{54}$

Get a numerical approximation of this exact result:

vpa(F_int)

ans =  $- 0.20200778585035447453044423341349$