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Math Help - Does Mid-point or Trapezoid estimates a greater value?

  1. #1
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    Does Mid-point or Trapezoid estimates a greater value?

    If a function f(x) is increasing and concaves up, odes mid-point or trapezoid approximation gives a greater value for approximation?
    I think it's mid-point approximation but my book says trapezoid.
    Can someone explain this please?
    Thanks!
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  2. #2
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    Re: Does Mid-point or Trapezoid estimates a greater value?

    Hey LLLLLL.

    You might want to draw an example and consider that the trapezoidal approximation includes the triangle portion in addition to the vertical bar that is the mid-point method.

    Get an idea visually by drawing a few examples and looking at the differences between the two area of integration graphically.
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  3. #3
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    Re: Does Mid-point or Trapezoid estimates a greater value?

    Hi,
    Your text is right, the trapezoid sum is greater than the midpoint sum. Moreover, it is not necessary that f be increasing, only that f is concave up. Here's the formal verification:

    Does Mid-point or Trapezoid estimates a greater value?-mhfcalc7.png
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