Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

12  The diagram shows the curve with equation y = x^2 (3 −  2 x − x^2 ). P and Q are

points on the curve with co-ordinates (−2, 12) and (1, 0) respectively.

(i) Find

d

d

y

x.

(ii) Find the equation of the line PQ.

(iii) Prove that the line PQ is a tangent to the curve at both P and Q.

(iv) Find the area of the region bounded by the line PQ and that part of the

curve for which − 2   x  1.   

[MEI]

13  The diagram shows the graph of y =  4 x − x^3. The point A has co-ordinates

(2, 0).

(i) Find d

d

y

x

.

Then find the equation of the tangent to the curve at A.

(ii) The tangent at A meets the curve again at the point B.

Show that the x co-ordinate of B satisfies the equation x^3 −   12 x +  16   = 0.

Find the co-ordinates of B.

(iii) Calculate the area of the shaded region between the straight line AB and

the curve.  

[MEI]

x

P y

Q

x

y

A

O

B

2