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lecture33hout

# lecture33hout - MATH 006 Calculus and Linear...

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MATH 006 Calculus and Linear Algebra (Lecture 33) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 14

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Outline 1 Area between two curves 2 An Application - Gini Index Albert Ku (HKUST) MATH 006 2 / 14
Area between two curves Area between two curves Consider the area R bounded by y = g ( x ) and y = f ( x ), where g ( x ) f ( x ), for a x b : R = (area under g ( x )) - (area under f ( x )) = Z b a g ( x ) dx - Z b a f ( x ) dx = Z b a [ g ( x ) - f ( x )] dx Albert Ku (HKUST) MATH 006 3 / 14

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Area between two curves Example Find the area bounded by y = 6 x - x 2 and y = 0 for 1 x 4. Solution Let g ( x ) = 6 x - x 2 and f ( x ) = 0. We need to find the range of values of x such that g ( x ) f ( x ) i.e. 6 x - x 2 = x (6 - x ) 0. By the following sign chart, g ( x ) f ( x ) for 0 x 6 i.e. the graph of y = 6 x - x 2 always lies above the graph of y = 0 for 1 x 4. Therefore, the area is Z 4 1 (6 x - x 2 ) dx = (3 x 2 - x 3 3 ) 4 1 = 24 Albert Ku (HKUST) MATH 006 4 / 14
Area between two curves Example Find the area between the graph of y = x 2 - 2 x and the x -axis over the interval [ - 1 , 1] Solution Similar to previous example, we need to find the range of values of x such that the graph of y = x 2 - 2 x

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lecture33hout - MATH 006 Calculus and Linear...

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