Page 34 - Spotlight A+ SPM Additional Mathematics Form 4 & 5

Form
5 Additional Mathematics Chapter 3 Integration

2. The generated volume of a solid is formed from a (c) Volume of each cylinder, dV i
revolved of x-axis is as follow: = Base area of cylinder × Height of cylinder
(a) Rotate an area of shaded region completely = πx × dy
2
i
through 360° about the x-axis until its = πx dy
2
i
generate a solid, approximately a cylinder. (d) Total volume of n cylinders
= V + V + V + …V
y y 1 2 3 n
n
y = f(x) ≈ ∑ dV
CHAP. i = 1 i
3 x D E O x n
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2
≈ ∑ πx dy
i
i = 1
(b) Divide the solid into n vertical cylinders
with a thickness of dx. y
Radius of
Radius of
cylinder
cylinder
y
x
x (e) When the number of cylinders is sufficiently
large, that is n ˜ ∞, then dy ˜ 0. Hence, the
generated volume of the solid:
n b
lim ∑ πx dy = πx dy
(c) Volume of each cylinder, dV i dx ˜ 0 i = 1 i 2 ∫ a 2
= Base area of cylinder × Height of cylinder

= πy × dx
2
i
= πy dx Determining the generated volume of a
2
i
(d) Total volume of n cylinders region revolved at the x-axis or y-axis
= V + V + V + …V
1 2 3 n
n A Generated volume, V through x-axis
≈ ∑ dV
i = 1 i y
n y = f(x)
≈ ∑ πy dx
2
i = 1 i x
(e) When the number of cylinders is sufficiently a b O
large, that is n ˜ ∞, then dx ˜ 0. Hence, the
generated volume of the solid:
The generated volume of a region bounded by the
n
∫
b
2
2
lim ∑ πy dx = πy dx curve y = f(x) is revolved through 360° about the
i
dx ˜ 0 i = 1 a x-axis is given by:
3. The generated volume of a solid is formed from a V = π y dx
∫
b
2
revolved of y-axis is as follow: a
(a) Rotate an area of shaded region completely
through 360° about the y-axis until its Example 17
generate a solid, approximately a cylinder.
Find the generated volume, in terms of π, when
y y the shaded region in each diagram is revolved
through 360° about the x-axis.
D
(a)
y
y = f(x) 2
E y = —
x
x x
O O

(b) Divide the solid into n horizontal cylinders O 1 3 x
with a thickness of dy.
280 3.3.4 3.3.5
C03 Spotlight Add Math F5.indd 280 23/04/2021 10:57 AM