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Calculating Areas Between Curves: Step-by-Step Guide

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In this guide, we explore the process of calculating the area between two curves or between a curve and the x-axis. We will analyze functions such as y = f(x) and y = g(x) and find the area enclosed between their intersections over a defined interval [a, b]. We will demonstrate how to graph the functions, identify points of intersection, and apply integration to determine the areas. This includes both simple curves and cases with multiple intersection points, along with practical examples that illustrate the steps needed for successful calculations.

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Calculating Areas Between Curves: Step-by-Step Guide

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  1. y y y = f(x) y = g(x) x x a b a b Shaded Area = Shaded Area = Section 5.3: Finding the Total Area

  2. y= f(x) y= f(x) y= f(x) y = g(x) y = g(x) a a b b y = g(x) Area under f(x)= Area under g(x)= a b A) Area Between Two Curves in [a , b] Area between f(x)and g(x)

  3. y y = g(x) y y = f(x) y = 0 y = 0 a b x x a b B) Area Between a Curve and the x-Axis in [a , b] x-Axis is same as y = 0 Top Function is y = 0 Top Function is y = f(x) Bottom Function is y = g(x) Bottom Function is y = 0 Area under f(x) in[a , b]: Area under g(x) in[a , b]:

  4. y y = 2x- 1 y = x2 - 4 x b a C) Area Between Intersecting Curves Example: Find the area between the graph y= x2 - 4and y = 2x- 1 1) Graph both functions 2) Find the points of intersection by equating both functions: y = y x2 - 4 = 2x- 1 x2 - 2x- 3 = 0 (x+ 1)(x- 3) = 0 x = -1 , x = +3 3) Area Top Function is 2x- 1, Bottom Function is x2 - 4

  5. y y = x2 - 4x +3 x 1 3 y = 0 Example: Find the area between the graph y = x2 -4x + 3and the x-axis 1) Graph the function with the x-axis (or y = 0) 2) Find the points of intersection by equating both functions: y = y x2 -4x + 3= 0 (x- 1)(x- 3) = 0 x = 1 , x = 3 3) Area Top Function is y = 0, Bottom Function is y = x2 -4x + 3

  6. y= x2 – 4 y y = 0 3 -1 -2 2 x A2= 2.33 A1= 9 = 9 + 2.33 = 11.33 D)Area Between Curves With Multiple Points of Intersections (Crossing Curves) Example: Find the area enclosed by the x-axis, y = x2 - 4, x = -1 and x = 3 1) Graph the function y = x2 - 4 with the x-axis (or y = 0), Shade in the region between x = -1 and x = 3 2) Find the points of intersection by equating both functions: y = y or (x- 2)(x+ 2) = 0 x2 -4= 0 x = 2 , x = -2 y = x2 - 4 and the x-axis cross each others x = -2, x = 2 3) Area

  7. y = x3 y y = x x 1 -1 0 A2= 0.25 A1= 0.25 , the answer will be zero. Note: if you write Area = Example: Find the area enclosed by y = x3 and y = x 1) Graph the function y = x3 , and y = x 2) Find the points of intersection by equating both functions: y = y or x3 - x = 0 x3 = x x(x2 - 1) = 0 x(x- 1)(x + 1) = 0 x = 0 , x = -1 , x = 1 The graphs cross at x = -1, x = 0 and x = 1 3) Area = 0.5 + 0.25 = 0.25

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