Page 32 - Spotlight A+ SPM Additional Mathematics Form 4 & 5

Form
5
Chapter 3 Integration Additional Mathematics
Explaining relation between the limit of the y
sum of areas of rectangles and the area b
∫
under a curve y = f(x) A = y dx
a

1. The area under curve y = f(x) can be determined A
by integration. O a b x
y
y = f(x) CHAP.
2. For the value of the area bounded by the curve
and the x-axis, 3
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A (a) if the region above the x-axis, then the
integral value is positive.
x
O a b
y y
2. Dividing the area under curve between x = a and
x = b into a few rectangles, dL.
y
y = f(x)
x
A A
y x x
2 O O
A y y A = y x

A 3 3
y A 1 2
1 (b) if the region below the x-axis, then the
integral value is negative.
O
x
For each of the following rectangles, y y
b – a
(a) the width is dx = , where n is the
n x x
number of rectangles. O A O A
(b) the height of the rectangle can be obtained
from the function of the curve, y .
i
Area of each rectangle, dA = Height × Width
i
≈ y × dx (c) if the region bounded by below the x-axis
i
≈ y dx and also above the x-axis, then the area of
i the region should be determined separately.
Hence, the total area of the n triangles,
≈ dA + dA + dA + … + dA n y
2
1
3
n
≈ ∑ dA
i = 1 i A
n x
≈ ∑ y dx O a b c
i = 1 i B
3. As the width of the strips of rectangle is become
thinner, the width of each rectangle, dx becomes
narrower and approaching to zero dx ˜ 0. Area of shaded region
Hence, = Area of A + Area of B
∫
∫
c
b
Area under the curve = lim ∑ ydx = y dx + y dx 
dx ˜ 0 a b
∫
b
= y dx
a
Determining the area of a region BRILLIANT Tips
A Area of a region between the curve and the The negative sign only indicates that the region is
x-axis below of x-axis. Therefore, negative signs can be
eleminated by using modulus, |a|.
1. The area under the curve bounded by x-axis = a
and x-axis = b given by:
3.3.2 275

C03 Spotlight Add Math F5.indd 275 23/04/2021 10:57 AM