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)F ind 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 360Â ° 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 360Â ° about the x-axis. The volume of the solid formed is 32?. Find the value of a. (Total 7 marks)