Question

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

Answer 1

Estimating Definite Integral by Euler's Method

Example

Use Euler's Method to approximate the definite integral

`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`

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I hope that this was helpful.