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Area Approximation - Mathematics Form 4

Estimation of the area under a curve

The information in this section provide a basis of how to estimate the area under a curve using regular shapes like a rectangle and a trapezium.


AREA APPROXIMATION
TRAPEZIUM AND MID-ORDINATE RULES

The knowledge of area helps in planning e.g. of towns, markets and cities to name but a few. Also in production for instance in farming farmers are able to predict or apportion the quantity of products expected in a given area. In finding area we involve regular (triangles, rectangular and trapezium) and irregular shapes (lakes, forests, plantations).

In this lesson we are going to derive and apply the trapezium and mid-ordinate rules in estimating area under a curve.

Deriving the trapezium rule
To estimate area under a curve y = f(x) between x = a and x = b, in the figure below, you divide the area into n strips of equal width h as shown, where y0
, y1
, y2
,---yn
are known as ordinates and h is the width of the trapezium.


The total estimated area under the curve y = f(x) between x = a and x = b is done by finding the sum of the trapezia.


NOTE:


The trapezia have the same width.
Therefore, the area under the curve y = f(x), between x = a and y = b is equal to
Area of trapezium A = 1/2 X h x (y
0
+ y1
)
Area of trapezium B = 1/2X h x (y
1
+ y2
)
Area of trapezium C = 1/2 X h x (y
2
+ y3
)
Area of trapezium D
= 1/2 X h x (y
3
+ y4
)


Area of trapezium E = 1/2X h x (y4
+ y5
)

Example 1
The speed of a train recorded at 10 seconds intervals as shown in the table below

(a)Draw a speed-time graph for the data given in the table.
(b)Use the trapezium rule to estimate the distance travelled
by the train in 50 seconds by dividing the region into five trapezia.
(c)Calculate the average speed of the train to the nearest kilometer per hour for the whole journey.


Solutions
(a) Draw the graph of time against speed.

Mid-ordinate Rule
The area under a curve can also be estimated using a number of rectangles of equal widths. Rather than use the end ordinates as the case in the trapezium method, this method uses the mid-ordinate of each strip. The height (h) of each rectangle is the ordinate of the curve at the mid-point of the interval.


The mid-ordinates show the heights of the rectangles respectively. The rectangles have equal width (h) which is found by dividing the length of the base by the number of strips. This is called the mid-ordinate method.


Deriving the mid-ordinate rule
Consider the figure below in which the area under the curve is to be estimated.
The estimated area under the curve is equal to the sum of the area of the rectangles.


Area of each rectangle = width of interval (h) X length of the mid-ordinate.
Total estimated area = (h X y
1
) + (h X y2
) + (h X y3
) + (h X y4
) + (h X y5
) + (h X y6
)
= h(y
1
+ y2
+ y3
+ y4
+ y5
+ y6
)
In general
The total estimated area under a curve is equal to
= h(y
1
+ y2
+ y3
+ . . . . + yn
)
This is known as the mid-ordinate rule.


Example 2
Estimate the area enclosed by the graph of the function y =x2
+ 4, the x-axis and the y-axis and line
x = 6. Using the mid-ordinate rule with five strips
.
Solution
Make a table of values for x and y.



The estimated area under the curve using mid-ordinate rule is
A = h (y
1
+ y2
+ y3
+ y4
+ y5
+ y6
)
Where
h = 1, y
1
= 4.25, y2
= 6.25, y3
= 10.25, y4
= 16.25, y5
= 24.25, y6
= 34.25 .
Area = 1 (4.25 + 6.25 + 10.25 + 16.25 + 24.25 + 34.25 )
Area = 95.5 square units.


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