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

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

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

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

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

y x Example: Find the area between the graph y = x 2 - 4x + 3 and 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 x = 1, x = 3 y = x 2 - 4x +3 x 2 - 4x + 3 = 0 (x - 1)(x - 3) = 0 1 3 3) Area Top Function is y = 0, Bottom Function is y = x 2 - 4x + 3 y = 0

y x Example: Find the area enclosed by the x-axis, y = x 2 - 4, x = -1 and x = 3 D) Area Between Curves With Multiple Points of Intersections (Crossing Curves) 1) Graph the function y = x 2 - 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 x = 2, x = -2 x 2 - 4 = 0 (x - 2)(x + 2) = 0 3) Area y = x 2 - 4 and the x-axis cross each others x = -2, x = 2 y= x 2 – 4 3 -2 2 y = 0 or = 9+ 2.33 = 11.33 A 1 = 9 A 2 = 2.33

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

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