# How do I integrate with Euler's method by hand?

Nov 13, 2014

Estimating Definite Integral by Euler's Method

Example

Use Euler's Method to approximate the definite integral

${\int}_{- 1}^{2} \left(4 - {x}^{2}\right) \mathrm{dx}$.

For simplicity, let us use the step size $\Delta x = 1$.

Let

$I \left(t\right) = {\int}_{- 1}^{t} \left(4 - {x}^{2}\right) \mathrm{dx}$.

So, we wish to approximate

$I \left(2\right) = {\int}_{- 1}^{2} \left(4 - {x}^{2}\right) \mathrm{dx}$

Note that by Fundamental Theorem of Calculus I,

$I ' \left(t\right) = 4 - {t}^{2}$

Now, let us start approximating.

$I \left(- 1\right) = \setminus {\int}_{- 1}^{- 1} \left(4 - {x}^{2}\right) \mathrm{dx} = 0$

By linear approximation,

$I \left(0\right) \approx I \left(- 1\right) + I ' \left(- 1\right) \cdot \Delta x = 0 + 3 \cdot 1 = 3$

$I \left(1\right) \approx I \left(0\right) + I ' \left(0\right) \cdot \Delta x \approx 3 + 4 \cdot 1 = 7$

$I \left(2\right) \approx I \left(1\right) + I ' \left(1\right) \cdot \Delta x \approx 7 + 3 \cdot 1 = 10$

Hence,

$I \left(2\right) = {\int}_{- 1}^{2} \left(4 - {x}^{2}\right) \mathrm{dx} \approx 10$

I hope that this was helpful.