Integration Using the Trapezoidal Rule

Key Questions

  • How does the trapezoidal rule work?

    Let us approximate the definite integral

    #int_a^b f(x)dx#

    by Trapezoid Rule #T_n#.

    First, split the interval #[a,b]# into #n# equal subintervals:

    #[x_0,x_1], [x_1,x_2],[x_2,x_3],...,[x_{n-1},x_{n}]#,

    where #a=x_0 < x_1 < x_2< cdots < x_n=b#.

    Trapezoid Rule #T_n# can be found by

    #T_n=[f(x_0)+2f(x_1)+2f(x_2)+cdots+2f(x_{n-1})+f(x_n)]{b-a}/{2n}#.

    I hope that this was helpful.

    Wataru · · Oct 17 2014
  • How do you Use the trapezoidal rule with four equal subdivisions to approximate a definite integral?

    First split the interval #[a,b]# into 4 equal subintervals:

    #[x_0,x_1],[x_1,x_2],[x_2,x_3]#, and #[x_3,x_4]#.

    (Note: #x_0=a# and #x_4=b#)

    The definite integral

    #int_a^b f(x)dx#

    can be approximated by

    #T_4=[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+f(x_4)]cdot{Delta x}/{2}#,

    where #Delta x={b-a}/4#.

    I hope that this is helpful.

    Wataru · 1 · Sep 28 2014
  • What is Integration Using the Trapezoidal Rule?

    Let us divide the interval #[a,b]# into n subintervals of equal lengths.

    #[a,b] to {[x_0,x_1], [x_1,x_2],[x_2,x_3],...,[x_{n-1},x_n]}#,

    where #a=x_0 < x_1 < x_2 < cdots < x_n=b#.

    We can approximate the definite integral

    #int_a^b f(x)dx#

    by Trapezoid Rule

    #T_n=[f(x_0)+2f(x_1)+2f(x_2)+cdots2f(x_{n-1})+f(x_n)]{b-a}/{2n}#

    Wataru · · Sep 22 2014

Questions

Methods of Approximating Integrals