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

Question

2191 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-11-13T22:50:09

Estimating Definite Integral by Euler's Method

Example

Use Euler's Method to approximate the definite integral

$\int_{- 1}^{2} (4 - x^{2}) d x$ $int_{-1}^2(4-x^2)dx$ .

For simplicity, let us use the step size $Deltax=1$ .

Let

$I(t)=int_{-1}^t(4-x^2)dx$ .

So, we wish to approximate

$I(2)=int_{-1}^2(4-x^2)dx$

Note that by Fundamental Theorem of Calculus I,

$I'(t)=4-t^2$

Now, let us start approximating.

$I(-1)=\int_{-1}^{-1}(4-x^2)dx=0$

By linear approximation,

$I(0) approx I(-1)+I'(-1)cdot Delta x=0+3cdot1=3$

$I(1) approx I(0)+I'(0)cdot Delta x approx3+4cdot1=7$

$I(2) approx I(1)+I'(1)cdot Delta x approx 7+3cdot1=10$

Hence,

$I(2)=int_{-1}^2(4-x^2)dx approx 10$

I hope that this was helpful.