Integration Using Euler's Method

Key Questions

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

    Estimating Definite Integral by Euler's Method

    Example

    Use Euler's Method to approximate the definite integral

    int_{-1}^2(4-x^2)dx∫2−1(4−x2)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.

    Wataru · 2 · Nov 13 2014
  • How do I integrate with Euler's method with a calculator or computer?

    To approximate an integral like \int_{a}^{b}f(x)\ dx with Euler's method, you first have to realize, by the Fundamental Theorem of Calculus, that this is the same as calculating F(b)-F(a), where F'(x)=f(x) for all x\in [a,b]. Also note that you can take F(a)=0 and just calculate F(b).

    In other words, since Euler's method is a way of approximating solutions of initial-value problems for first-order differential equations, we want to calculate \int_{a}^{b}f(x)\ dx=F(b) by approximating the unique solution of dy/dx=f(x), y(a)=0 at x=b.

    This can be done with an "iteration scheme". Pick a positive integer n to be the number of steps of Euler's method you want to use and then let Delta x=(b-a)/n. Once this is done, let x_{0}=a and y_{0}=0 and use the recursive equations x_{k+1}=x_{k}+Delta x, y_{k+1}=y_{k}+Delta y=y_{k}+f(x_{k})\cdot Delta x to generate a sequence of n+1 points (x_{0},y_{0}), (x_{1}, y_{1}), \ldots (x_{n},y_{n}) that approximate the unique solution of the initial-value problem.

    The final result is that \int_{a}^{b}f(x)\ dx=F(b)\approx y_{n}.

    This can be implemented fairly easily on a calculator or computer, though you'd have to be somewhat experienced with such programming.

    As an example, suppose that you want to estimate \int_{0}^{3}x^{2}\ dx (which we already know is 9). The relevant initial-value problem is dy/dx=f(x)=x^2, y(0)=0 and we want to approximate y(3). Let's choose n=4 so that Delta x=\frac{3}{4}=0.75. Then x_{0}=0, x_{1}=0.75, x_{2}=1.5, x_{3}=2.25, and x_{4}=3. Also y_{1}=y_{0}+f(0)\cdot 0.75=0+0=0, y_{2)=y_{1}+f(0.75)\cdot 0.75=0+0.5625\cdot 0.75=0.421875, y_{3}=y_{2}+f(1.5)\cdot 0.75=0.421875+2.25\cdot 0.75=2.109375, and y_{4}=y_{3}+f(2.25)\cdot 0.75=2.109375+3.796875=5.90625 as our approximate answer for the integral.

    This would get to be a better approximation (though very slowly) as n increases (and Delta x=(b-a)/n decreases). For instance, if n=100 and Delta x=3/100=0.03, the approximation for the integral is 8.86545.

    Bill K. · 1 · May 19 2015
  • What is Integration Using Euler's Method?

    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.

    Wataru · 1 · Nov 6 2014

Questions

Methods of Approximating Integrals