1. The area of the enclosed region shown in the diagram is defined by
y ³ x2 + 2, y £ ax + 2, where a > 0.
This region is rotated 360° about the xaxis to form a solid of revolution. Find, in terms of a, the volume of this solid of revolution.
Working:



Answer:
…………………………………………..

(Total 4 marks)
2. Using the substitution u = x + 1, or otherwise, find the integral
dx.
Working:



Answer:
…………………………………………..

(Total 4 marks)
3. When air is released from an inflated balloon it is found that the rate of decrease of the volume of the balloon is proportional to the volume of the balloon. This can be represented by the differential equation = – kv, where v is the volume, t is the time and k is the constant of proportionality.
(a) If the initial volume of the balloon is v0, find an expression, in terms of k, for the volume of the balloon at time t.
(b) Find an expression, in terms of k, for the time when the volume is
Working:



Answers:
(a) …………………………………………..
(b) ……………………………………..........

(Total 4 marks)
4. Consider the function f : x x – x2 for –1 £ x £ k, where 1 < k £ 3.
(a) Sketch the graph of the function f.
(3)
(b) Find the total finite area enclosed by the graph of f, the xaxis and the line x = k.
(4)
(Total 7 marks)
5. The area between the graph of y = ex and the xaxis from x = 0 to x = k (k > 0) is rotated through 360° about the xaxis. Find, in terms of k and e, the volume of the solid generated.
Working:



Answer:
....……………………………………..........

(Total 4 marks)
6. Find the real number k > 1 for which dx = .
Working:



Answer:
....……………………………………..........

(Total 4 marks)
7. In the diagram, PTQ is an arc of the parabola y = a2 – x2, where a is a positive constant, and PQRS is a rectangle. The area of the rectangle PQRS is equal to the area between the arc PTQ of the parabola and the xaxis.
Find, in terms of a, the dimensions of the rectangle.
Working:



Answer:
....……………………………………..........

(Total 4 marks)
8. Consider the function fk (x) = , where k Î
(a) Find the derivative of fk (x), x > 0.
(2)
(b) Find the interval over which f0 (x) is increasing.
The graph of the function fk (x) is shown below.
(2)
(c) (i) Show that the stationary point of fk (x) is at x = ek–1.
(ii) One xintercept is at (0, 0). Find the coordinates of the other xintercept.
(4)
(d) Find the area enclosed by the curve and the xaxis.
(5)
(e) Find the equation of the tangent to the curve at A.
(2)
(f) Show that the area of the triangular region created by the tangent and the
coordinate axes is twice the area enclosed by the curve and the xaxis.
(2)
(g) Show that the xintercepts of fk (x) for consecutive values of k form a geometric sequence.
(3)
(Total 20 marks)
9. Find the values of a > 0, such that dx = 0.22.
Working:



Answer:
..................................................................

(Total 3 marks)
10. Let f (x) = ln x5 – 3x2, –0.5 < x < 2, x ¹ a, x ¹ b; (a, b are values of x for which f (x) is not defined).
(a) (i) Sketch the graph of f (x), indicating on your sketch the number of zeros of f (x). Show also the position of any asymptotes.
(2)
(ii) Find all the zeros of f (x), (that is, solve f (x) = 0).
(3)
(b) Find the exact values of a and b.
(3)
(c) Find f (x), and indicate clearly where f¢ (x) is not defined.
(3)
(d) Find the exact value of the xcoordinate of the local maximum of f (x), for 0 < x < 1.5. (You may assume that there is no point of inflexion.)
(3)
(e) Write down the definite integral that represents the area of the region enclosed by f (x) and the xaxis. (Do not evaluate the integral.)
(2)
(Total 16 marks)
11. Calculate the area bounded by the graph of y = x sin (x2) and the xaxis, between x = 0 and the smallest positive xintercept.
Working:



Answer:
..................................................................

(Total 3 marks)
