Integration196P1^
6
EXERCISE 6C 1 Sketch each of these curves and find the area between the curve and the x axis
between the given bounds.
(i) y = x^3 between x = − 3 and x = 0.
(ii) y = x^2 − 4 between x = − 1 and x = 2.
(iii) y = x^5 − 2 between x = − 1 and x = 0.
(iv) y = 3 x^2 − 4 x between x = 0 and x = 1.
(v) y = x^4 − x^2 between x = − 1 and x = 1.
(vi) y = 4 x^3 − 3 x^2 between x = − 1 and x = 0.5.
(vii) y = x^5 − x^3 between x = − 1 and x = 1.
(viii) y = x^2 − x − 2 between x = − 2 and x = 3.
(ix) y = x^3 + x^2 − 2 x between x = − 3 and x = 2.
(x) y = x^3 + x^2 between x = − 2 and x = 2.
2 The diagram shows a sketch of part of the curve with equation y = 5 x^4 − x^5.(i) Find
d
dy
x.
Calculate the co-ordinates of the stationary points.
(ii) Calculate the area of the shaded region enclosed by the curve and the x axis.(iii) Evaluate (^) ∫
6
0 x
(^4) (5 − x) dx and comment on your result.
[MEI]
3 (i) (a) Find (^1)
4
(^121)
∫ () 3 −^8
x
d.x
dx.
(b) Find (^1)
2
1
3
(^18)
∫ ()x − dd.xx.
(ii) Hence find the total area of the regions bounded by the curve y
x
=−^138 ,
the lines x = 14 and x = 1 and the x axis.4 (i) (a) Find (^) ∫^4022 xx()− d.x
(b) Find (^) ∫ 4922 xx()− d.x
(ii) Hence find the total area of the regions bounded by the curve
yx=− (^22) ()x , the line x = 9 and the x axis.
x
y