# Area and Volume

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 marks)

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