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Thread: Improving accuracy of trapezoid and simpsons rule in integration

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    Improving accuracy of trapezoid and simpsons rule in integration

    So far this is what i got,
    use trapezoid rule for linear problems, simpsons rule for quadratic polynomial (parabolas)
    n value can be odd or even for trapezoid (better if its odd)
    n value should be even for simpsons rule
    to improve accuracy for both rules, increase number of n.

    are these correct? Other than increasing number of n for both rules, what are some other ways to improve the accuracy for these two rules?

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    Re: Improving accuracy of trapezoid and simpsons rule in integration

    Quote Originally Posted by dgt20 View Post
    use trapezoid rule for linear problems, simpsons rule for quadratic polynomial (parabolas)
    Yep.

    The Trapezoidal Rule is exact when integrating any linear function.

    Simpson's Rule is exact when integrating any quadratic or cubic. Generally it's also more accurate with integrating any curvy function.

    Quote Originally Posted by dgt20 View Post
    n value can be odd or even for trapezoid (better if its odd)
    It shouldn't matter how many subintervals there are. Because you're just approximating the curve with trapeziums, having 5 trapeziums vs. having 6 shouldn't make a difference (aside from the increased accuracy you get with more function values).

    Quote Originally Posted by dgt20 View Post
    n value should be even for simpsons rule
    Yes, it must be even otherwise Simpson's Rule won't work properly, since it uses three points (two subintervals) to generate the parabola that approximates the curve. So curves are always approximated two subintervals at a time.

    Quote Originally Posted by dgt20 View Post
    to improve accuracy for both rules, increase number of n.
    Yep. More subintervals means your trapeziums/parabolas will better approximate the function.

    Quote Originally Posted by dgt20 View Post
    Other than increasing number of n for both rules, what are some other ways to improve the accuracy for these two rules?
    Besides taking more subintervals, you can't really improve on the accuracy without changing the rule itself.
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