Area and Volume
Previous exam questions on area between functions and volumes of solids.
1. Let f(x) = cos(x2) and g(x) = ex, for –1.5 ≤ x ≤ 0.5. Find the area of the region enclosed by the graphs of f and g.
(Total 6 marks)
2. Let f(x) = Aekx + 3. Part of the graph of f is shown below. The y-intercept is at (0, 13). (a) Show that A =10.
(2)
(b) Given that f(15) = 3.49 (correct to 3 significant figures), find the value of k.
(3)
(c) (i) Using your value of k, find f′(x).
(ii) Hence, explain why f is a decreasing function.
(iii) Write down the equation of the horizontal asymptote of the graph f.
(5)
Let g(x) = –x2 + 12x – 24.
(d) Find the area enclosed by the graphs of f and g.
(6)
(Total 16 marks)
3. The following diagram shows the graphs of f (x) = ln (3x – 2) + 1 and g (x) = – 4 cos (0.5x) + 2, for 1 £ x £ 10.
(a) Let A be the area of the region enclosed by the curves of f and g.
(i) Find an expression for A.
(ii) Calculate the value of A.
(6)
(b) (i) Find f ′ (x).
(ii) Find g′ (x).
(4)
(c) There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x.
(4)
(Total 14 marks)
4. The graph of f(x) = , for –2 ≤ x ≤ 2, is shown below. The region enclosed by the curve of f and the x-axis is rotated 360° about the x-axis.
Find the volume of the solid formed.
(Total 6 marks)
5. The graph of y = between x = 0 and x = a is rotated 360° about the x-axis.
The volume of the solid formed is 32π. Find the value of a.
(Total 7
1. Let f(x) = cos(x2) and g(x) = ex, for –1.5 ≤ x ≤ 0.5. Find the area of the region enclosed by the graphs of f and g.
(Total 6 marks)
2. Let f(x) = Aekx + 3. Part of the graph of f is shown below. The y-intercept is at (0, 13). (a) Show that A =10.
(2)
(b) Given that f(15) = 3.49 (correct to 3 significant figures), find the value of k.
(3)
(c) (i) Using your value of k, find f′(x).
(ii) Hence, explain why f is a decreasing function.
(iii) Write down the equation of the horizontal asymptote of the graph f.
(5)
Let g(x) = –x2 + 12x – 24.
(d) Find the area enclosed by the graphs of f and g.
(6)
(Total 16 marks)
3. The following diagram shows the graphs of f (x) = ln (3x – 2) + 1 and g (x) = – 4 cos (0.5x) + 2, for 1 £ x £ 10.
(a) Let A be the area of the region enclosed by the curves of f and g.
(i) Find an expression for A.
(ii) Calculate the value of A.
(6)
(b) (i) Find f ′ (x).
(ii) Find g′ (x).
(4)
(c) There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x.
(4)
(Total 14 marks)
4. The graph of f(x) = , for –2 ≤ x ≤ 2, is shown below. The region enclosed by the curve of f and the x-axis is rotated 360° about the x-axis.
Find the volume of the solid formed.
(Total 6 marks)
5. The graph of y = between x = 0 and x = a is rotated 360° about the x-axis.
The volume of the solid formed is 32π. Find the value of a.
(Total 7