Numerical integration in python

numerical integration is a powerful technique used to approximate the integral of a function when an analytical solution is difficult or impossible to obtain. in python, there are several libraries available for numerical integration, with numpy and scipy being two of the most commonly used.

### overview of numerical integration methods

1. **rectangular method (midpoint rule)**: approximates the area under the curve by dividing it into rectangles.
2. **trapezoidal rule**: approximates the area under the curve by dividing it into trapezoids.
3. **simpson’s rule**: uses parabolic segments to approximate the area under the curve.
4. **adaptive quadrature**: adjusts the integration process based on the function’s behavior.

### using scipy for numerical integration

- `quad`: for single integrals.
- `dblquad`: for double integrals.
- `tplquad`: for triple integrals.
- `simps`: for simpson's rule.
- `trapz`: for the trapezoidal rule.

let's consider a simple example of integrating the function \( f(x) = x^2 \) from 0 to 1.

#### step 1: install scipy

if you haven't installed scipy yet, you can do so using pip:

#### step 2: write the code

### explanation of the code

1. **function definition**: we define the function \( f(x) = x^2 \) to integrate.
2. **integration limits**: we set the limits of integration from 0 to 1.
4. **results display**: the result and error are printed to the console.
5. **visualization**: we plot the function and shade the area under the curve between the limits to visualize the integral.

### conclusion

numerical integration is a vital tool in computa ...

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