Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

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

Show that the definite integral for on is 0.

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

ans =

To maximize for , first, define the symbolic variables and assume that :

syms a x assume(a >= 0);

Then, define the function to maximize:

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

F =

Note the special case here for . To make computations easier, use `assumeAlso`

to ignore this possibility (and later check that is not the maximum):

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

F =

Create a plot of to check its shape:

fplot(F,[0 10])

Use `diff`

to find the derivative of with respect to :

Fa = diff(F,a)

Fa =

The zeros of are the local extrema of :

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

The maximum is between 1 and 2. Use `vpasolve`

to find an approximation of the zero of in this interval:

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

a_max =

Use `subs`

to get the maximal value of the integral:

F_max = subs(F,a,a_max)

F_max =

The result still contains exact numbers and . Use `vpa`

to replace these by numerical approximations:

vpa(F_max)

ans =

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

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

ans =

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 =

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 , where the on the right-hand-side denotes a scalar product.

Use this definition to compute the line integral for from

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

F_int =

Get a numerical approximation of this exact result:

vpa(F_int)

ans =