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Date/page #: _____________ Areas Between Curves Obj 10.7 Entry Slip: Yes, you need to use a calculator, but you also need to show how the problem gets set up. No credit for just an answer. 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) Date/page #: _____________ Areas Between Curves Obj 10.7 Do Now: Objectives: Do Now: Area between curves: Find the area of the shaded region: What did you have to do to get these areas? Notes/Guided Practice: The same logic applies to finding areas between curves. Find the area between the following two curves and the x-axis on the interval: y = x2 y = 2x – x2 Shade the area bounded the two parabolas y = x2 and y = 2x – x2 and find the area of that region. Graph of y = x2 and y = 2x – x2 To get area between the curves, what do we need to do? What if we want the area between ANY two functions, say f(x) and g(x)? What is a general formula we can use to find any area between two curves? Are there any restrictions that we need to put on the two functions? For the parabola problem defined above: f(x) = g(x) = What is the integral for this problem? Solve the integral to find the area of the region enclosed by the curves. a= b= Find the area bounded by the x-axis and y = x 2 - 2x . First, without a calculator, lets get an idea of what the graph will look like. What are the x intercepts? What is the y intercept? What is the vertex? Sketch the graph and shade the area you are being asked to find. f(x) = g(x) = a= b= Set up and solve the interval. Find the area of the region bounded above by y = ex, bounded below by y = x and bounded on the sides by x = 0 and x = 1. Find the area of the region enclosed by y = x + 2 and y = x2 + x – 2. Find the total area of the regions contained by y = f(x) and the x-axis for f(x) = x3 + 2x2 – 3x. What was different about the last example from the other examples? When this happens, do we need to do anything special? If yes, how can address this situation? Find the area of the region bounded by the curves y = sin x and y = cos x from x = 0 to x = /2. Date/page #: _____________ Areas Between Curves Obj 10.7 Warm UP: Exercise 19B 1: choose 4 Choose 2 or 4 Choose 3 or 5 6 8 10 – 12 14 – 16 Date/page #: _____________ Areas Between Curves Obj 10.7 To Be Handed In On: ________________________________ Y59. (Yes) 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) N14. Consider the curve y = xex and the line y = kx, k (a) . Let k = 0. (i) Show that the curve and the line intersect once. (ii) Find the angle between the tangent to the curve and the line at the point of intersection. (5) (b) Let k = 1. Show that the line is a tangent to the curve. (3) (c) (i) Find the values of k for which the curve y = xex and the line y = kx meet in two distinct points. (ii) Write down the coordinates of the points of intersection. (iii) Write down an integral representing the area of the region A enclosed by the curve and the line. (iv) Hence, given that 0 < k < 1, show that A < 1. (15) (Total 23 marks) N60. Let f (x) = ex (1 – x2). (a) Show that f ′ (x) = ex (1 – 2x – x2). (3) Part of the graph of y = f (x), for – 6 x 2, is shown below. The x-coordinates of the local minimum and maximum points are r and s respectively. (b) Write down the equation of the horizontal asymptote. (1) (c) Write down the value of r and of s. (4) (d) Let L be the normal to the curve of f at P(0, 1). Show that L has equation x + y = 1. (4) (e) Let R be the region enclosed by the curve y = f (x) and the line L. (i) Find an expression for the area of R. (ii) Calculate the area of R. (5) (Total 17 marks)