1. 5. 7a Numerical Integration. Trapezoidal sums
5.7a Numerical Integration. Trapezoidal sums. Trapezoidal Rule (optional).
Rita Korsunsky
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2. Trapezoidal approximation
With the Trapezoidal approximation,
instead of approximating area by using rectangles (as you do with the left, right, and midpoint Riemann sum methods),
you approximate area with trapezoids.

3. Example 1 3 5 7 10 11 20 22

4. Example 2
(seconds) 20 24 25 28 35 40
(meters/ seconds) 4 6 3 8 12 12 10 8 6 4 2
…20 24 28 32 36 40

5. Example 3

6. Example 4

7. Trapezoidal Rule (not tested on AP Exam)
If all subintervals are of equal length, then we use this formula:
Trapezoidal Rule
Conditions: Let f be continuous on [a,b]. If a regular partition of [a,b] is determined by a = ao, a1, …, an = b
f (x) a b a0 a2 a3 a4 a5 a6 a1 x
Trapezoidal Rule gives the sum of the areas of the trapezoids under the curve.
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8. Proof of Trapezoidal Rule
1. Area of each individual trapezoid:
f(x) x
From the diagram, b1 b2 a0 h a1
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9. Trapezoidal Rule
2.Since we divide the curve equally in the x-direction, and since there are in total of n blocks. a1-a0=…=an - an-1 will always be (b-a)/n.
Let’s find and add the areas of 3 trapezoids:
f(x) x
a0 = a a1 a2
a3 = b
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10. Trapezoidal Rule Yielding the form… link Yielding the general form:
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11. Applications of the Trapezoidal Rule
Yielding the general form:
Applications of the Trapezoidal Rule
The values of the function are given in the table below. x 1 2 3 4 5 6 7 8 9 10
f(x) 20
19.5 18
15.5 12
7.5
-4.5
-12
-20.5
-30
Rita Korsunsky