# How to write and evaluate definite integrals (in Python, using SymPy)

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The area under a curve can be computed using a definite integral. To compute the area above the $x$ axis and under $f(x)$, from $x=a$ to $x=b$, we write

$\int_a^b f(x)\;dx.$

How can we write and evaluate definite integrals using software?

## Solution

This answer assumes you have imported SymPy as follows.

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from sympy import *                   # load all math functions
init_printing( use_latex='mathjax' )  # use pretty math output


Letâ€™s compute the area under $\sin x$ from $x=0$ to $x=\pi$.

We use the same technique as in how to write and evaluate indefinite integrals, except that we add the lower and upper bounds together with $x$, as shown below.

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var( 'x' )
formula = sin(x)
Integral( formula, (x,0,pi) )


$\displaystyle \int\limits_{0}^{\pi} \sin{\left(x \right)}\, dx$

The above code just displays the definite integral. To evaluate it, use the integrate command.

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integrate( formula, (x,0,pi) )


$\displaystyle 2$