MATHEMATICS MATHEMATICS FORM 5FORM 5

CHAPTER 6CHAPTER 6

GRADIENT AND AREA GRADIENT AND AREA UNDER A GRAPHUNDER A GRAPH

6.2 AREA UNDER 6.2 AREA UNDER A GRAPHA GRAPH

LEARNING OUTCOMES:LEARNING OUTCOMES:

By the end of this lesson, students will be By the end of this lesson, students will be able to:able to:

1. Find the area under a graph.1. Find the area under a graph.

2. determine the distance by finding the 2. determine the distance by finding the area under the following types of speed-area under the following types of speed-time graph;time graph;

a) a) v = kv = k

b) b) v = ktv = kt

c) c) v = kt + hv = kt + h

d) a combination of the aboved) a combination of the above

6.2 (B) Finding the Area 6.2 (B) Finding the Area Under a GraphUnder a Graph

Steps to find the area under a Steps to find the area under a graph:graph:

i) Draw one or more dashed lines i) Draw one or more dashed lines which are perpendicular to the which are perpendicular to the x-x-axis.axis.

ii) Recognize the shape for the ii) Recognize the shape for the different region.different region.

iii) Use suitable formulae to find the iii) Use suitable formulae to find the areas.areas.

Example 1Example 1

Find the area under the graphs:Find the area under the graphs:

6

y

x

8

0 6

y

x

8

0

6.2 (C) Determining the 6.2 (C) Determining the Distance from the Speed-Distance from the Speed-

Time GraphTime Grapha) When the particle travels at a a) When the particle travels at a

uniform speed:uniform speed:

v = kv = k

Speed (m/s)

Time (s)

30

0 30

b) When the speed starts from 0 and b) When the speed starts from 0 and changes at a fixed rate:changes at a fixed rate:

v = ktv = kt

Speed (m/s)

Time (s)

24

0 40

c) When the speed starts from a c) When the speed starts from a certain value and changes at a fixed certain value and changes at a fixed rate;rate;

v = kt + hv = kt + hSpeed (m/s)

Time (s)

40

020

20

d) A combination of the above.d) A combination of the above.

When the speed is not consistent, When the speed is not consistent, divide the area into several divide the area into several polygons. Find the area of each polygons. Find the area of each polygons and sum them up.polygons and sum them up.

Speed (m/s)

Time (s)

40

10 20 35

CBA

Summary Summary

Quantity represented by the area Quantity represented by the area under a graph is the product of the under a graph is the product of the quantity represented by the quantity represented by the horizontal axis and the quantity horizontal axis and the quantity represented by the vertical axis.represented by the vertical axis.

Area under a speed-time graph Area under a speed-time graph represents the distance traveledrepresents the distance traveled