Ch 5.5 Trapezoidal Rule Graphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy.

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