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How to Do the Trapezoidal Riemann Sum

Finding the area of the region under a curve requires the use of a Riemann sum called the trapezoidal rule. The Riemann sum process breaks up the region under the curve into trapezoids, finds the area of the trapezoids, then sums the areas together to approximate the area under the curve. The trapezoidal rule is especially accurate when solving for the areas under periodic functions, such as sine and cosine graphs. The result of a function solved by the trapezoidal rule is the same as finding the definite integral of that function.

Instructions

- 1
  Find the length of each interval by subtracting the final point of the interval from the initial point of the interval ("x) then dividing by the number of subintervals. For example, if you are using the trapezoidal rule on the interval (3, 8) with 10 subintervals, the equation becomes: "x = (8 - 3) / 10 = (5 / 10) = (1 / 2) = 0.5.
- 2
  Divide "x by 2. For example, ("x = (1 / 2) / 2 becomes ((0.5) / 2) = (1 / 4) = 0.25.
- 3
  Multiply this new value by the sum of the function f(x) at each subinterval. For example, if "x = 0.5, ("x / 2) = 0.25 and you wish to approximate the area of the integral (1 / x) on the interval (3, 8) with 10 subintervals, the trapezoidal rule "T" gives: T = (0.25) * ((1 / 3) + (2 / 3.5) + (2 / 4) + f(2 / 4.5) + (2 / 5) + (2 / 5.5) + (2 / 6) + (2 / 6.5) + (2 / 7) + (2 / 7.5) + (1 / 8)) becomes (0.25) * (3.93) = 0.98.

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