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What is Integration Using Euler's Method?

Answer 1

Charles Anctil

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#

I hope that this was helpful.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.