# Documentation

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## Integration

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

### Definite Integral

Show that the definite integral for on is 0.

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

### Definite Integrals in Maxima and Minima

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 = `

### 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 = `

### 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 , 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 = `