1. Trapezoidal Approximation
There are 3 basic approximation techniques that use rectangles:
Left Endpoints
Right Endpoints
Midpoints
If we connect the left and right endpoint for a subinterval with a line, we create a trapezoid…
Area of a trapezoid b h2 h1

2. An addition of several trapezoids in sequence shows that all endpoint function heights except the first left and the final right will be used in the calculation twice… A B C D x0 x1 x2 x3 x4
used in the calculation of area A and B
used in the calculation of area A
used in the calculation of area B and C
used in the calculation of area C and D
used in the calculation of area D

3. (using ‘n’ trapezoids)
Thus, for
(an approximation)
(using ‘n’ trapezoids)
where
and
ex: Use the Trapezoidal Rule with n = 5 to approximate
n = 5
a = 1
b = 2
x0 = a = 1
x1 = a + 1(.2) = 1.2
x2 = a + 2(.2) = 1.4
x3 = a + 3(.2) = 1.6
x4 = a + 4(.2) = 1.8
x5 = b = .2

4. Error estimation
Upper bounds for the error
Gives theoretical, worst-case scenario
Actual error may be less
Steps
Calculate f ’’(x)
Determine maximum value for f ’’(x) based on limits (call this ‘K’)

5. IT’S A TRAP!!…. EZOID…
IT’S A TRAP!!….