Section 5.3: Finding the Total Area

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Presentation transcript:

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

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

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

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

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

y = x2 - 4 and the x-axis cross each others x = -2, x = 2 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 y x 2) Find the points of intersection by equating both functions: y= x2 – 4 y = y 3 -1 y = 0 x2 - 4 = 0 or (x - 2)(x + 2) = 0 x = 2 , x = -2 -2 2 y = x2 - 4 and the x-axis cross each others x = -2, x = 2 A2 = 2.33 3) Area A1 = 9 = 9 + 2.33 = 11.33

The graphs cross at x = -1, x = 0 and x = 1 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 y x y = x3 x3 = x or x3 - x = 0 y = x x(x2 - 1) = 0 1 -1 x(x - 1)(x + 1) = 0 x = 0 , x = -1 , x = 1 The graphs cross at x = -1, x = 0 and x = 1 A2 = 0.25 3) Area A1 = 0.25 = 0.25 + 0.25 = 0.5 Note: if you write Area = , the answer will be zero.