Chapter(^6)
299
P1^
15 y = 6 x − x
2
2 +^2
16 (i) y = 4 x − 12 x^2 + 3
(ii) x + 2 y = 20
(iii) (7, 6.5)
Activity 6.1 (Page 183)
The bounds converge on the value
A = (^4513).
Activity 6.2 (Page 187)
(i) Area = 2 1 [3 + (b + 3)]b − 21 [3 + (a + 3)]a
= 21 [6b + b^2 − 6 a − a^2 ]
(ii) = (^) [b
2
2 +^3 b]^ −^ [
a^2
2 +^3 a]
= (^) [x
2
2 +^3 x]
b
a
(iii) (^) ∫ba(x + 3) dx = (^) [x
2
2 +^3 x]
b
a
Exercise 6B (Page 189)
1 (i) x^3 + c
(ii) x^5 + x^7 + c
(iii) 2 x^3 + 5 x + c
(iv) xx x xc
43 2
43 + + 2 + +
(v) x^11 + x^10 + c
(vi) x^3 + x^2 + x + c
(vii) x
3
3 +^5 x^ +^ c^
(viii) 5 x + c
(ix) 2 x^3 + 2 x^2 + c
(x) x
5
5 +^ x
(^3) + x (^2) + x + c
2 (i) − 103 x−^3 + c
(ii) x^2 + x−^3 + c
(iii) 2 x + x
4
4 −^
5
2 x
− (^2) + c
(iv) 2 x^3 + 7 x−^1 + c
(v) 4 x
(^54)
- c
(vi) − 31 x 3 + c
(vii) 23 x x + c
(viii) (^25)
x^5 - (^4) x + c
3 (i) 3
(ii) 9
(iii) 27
(iv) 12
(v) 12
(vi) 15
(vii) 114
(viii) 16
(ix) 2209
(x) 0
(xi) –105^34
(xii) 5
4 (i) 241
(ii) (^34)
(iii) 56
(iv) − (^223)
(v) (^1758)
(vi) (^1023)
5 (i) A: (2, 4); B: (3, 6)
(ii) 5
(iv) In this case the area is not a
trapezium since the top is
curved.
6 (i)
(ii) 213
7 (i)
(ii) − 2 x 2
(iii) 1023
8 (i)
(ii) 223 square units
9 2113 square units
10 (i)
(ii) 1823 square units
11 (i)
(ii) y = x^2
(iii) y = x^2 : area = 13 square units
y = x^3 : area = 14 square units
(iv) Expect ∫^21 x^3 dx ∫^21 x^2 dx,
since the curve y = x^3 is
above the curve y = x^2
between 1 and 2.
Confirmation: ∫ 12 x^3 dx = (^334)
and ∫ 12 x^2 dx = (^213)
12 (i)
O 1 2 x
y
y
–2 2
4
O x
– O 2 x
y
O 2 x
y
–1 1 3 4
O x
y y = x (^3) y = x 2
y
–1 –1O 1 2 x