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Ch 5.5 Trapezoidal RuleGraphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy
Approximation of Area Under a Curve
1 2
A more accurate approximation of areea under a curve
can be found by finding the area of trapezoids (rather
than LRAM, RRAM, MRAM).
hArea of a trapezoid = b + b
2
Example
2Approximate the area under the curve y = x , on the interval
1,2 , by dividing the curve into 4 equal lengths and using the
area of trapezoids.
Example
2Approximate the area under the curve y = x , on the interval
1,2 , by dividing the curve into 4 equal lengths and using the
area of trapezoids.
2 2 2
2 2 2
1 5 6 7Area = 1 + 2 + + + 4
8 4 4 4
1 110 = 5 + = 2.34375
8 8
Activity
n nn
Using the definitions, prove that, in general,
LRAM + RRAMTrap =
2
Activity
n nn
b x bn n
n a n a x
Using the definitions, prove that, in general,
LRAM + RRAMTrap =
2b a
Let h x = n
LRAM + RRAM 1= f x x + f x x
2 2
1 b a = f a + 2 f a x f a 2 x .
2 n
0 1 2 n 1 n
n
.. f b x +f b
h = y 2 y y ...y y
2 = Trap
Example
The table below records the outside temperature every hour from noon until midnight. What was the average temperature for the 12-hour period?
Time
Temp
N 1 2 3 4 5 6 7 8 9 10 11 M
63 65 66 68 70 69 68 68 65 64 62 58 55
Example
The table below records the outside temperature every hour from noon until midnight. What was the average temperature for the 12-hour period?
Time
Temp
N 1 2 3 4 5 6 7 8 9 10 11 M
63 65 66 68 70 69 68 68 65 64 62 58 55
1 1Avg = 63 2 65 66 68 70 68 68 68 65 64 62 58 55
12 2
1 = 782 65.17
12