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Application of Definite Integral

# Application of Definite Integral - Applications of Definite...

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Applications of Definite Integrals Advanced Level Pure Mathematics Advanced Level Pure Mathematics Applications of DeFnite Integrals Areas 2 Arc Length 8 Volumes of Solids of Revolution 11 Area of Surface of Revolution 13 Prepared by Mr. K. F. Ngai Page 1 8

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Applications of Definite Integrals Advanced Level Pure Mathematics Areas A Equations of Curves are represented in Rectangular Form Let A denote the area ( or total area) of the shaded region. Theorem The area enclosed by the graph of ) x ( f y , the x -axis and the lines a x and b x is equal to dx b a ) x ( f or dx b a y . Theorem The area enclosed by the graph of ) y ( g x , the x -axis and the lines c y and d y is equal to dy d c ) y ( g or dy d c x . Example Find the area enclosed by the graph of 2 x y , the axis and the lines 1 x and 3 x . Prepared by Mr. K. F. Ngai Page 2
Applications of Definite Integrals Advanced Level Pure Mathematics Solution Example Find the area bounded by the following curves. (a) Ellipse: 1 b y a x 2 2 2 2 . (b) Cycloid: ) t cos 1 ( a y ) t sin t ( a x , π 2 t 0 and x -axis. Example Find the area enclosed by the graph 1 x y 2 , the y -axis and the lines 2 y and 3 y . Prepared by Mr. K. F. Ngai Page 3

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Applications of Definite Integrals Advanced Level Pure Mathematics Example Given a conic 0 9 y 8 x 6 y 4 x : C 2 2 . (a) By completing squares and translation coordinate axes, transform the equation of C to standard form. What is this curve? (b) find the area of the region bounded by C . ( Ans: π 2 ) Solution B Equations of Curves are in parametric Form Prepared by Mr. K. F. Ngai Page 4
Applications of Definite Integrals Advanced Level Pure Mathematics It is known that the area between the curve ) x ( f y and the lines a x , b x and 0 y is given by dx b a y . If the equation of the curve is in parametric form ) t ( G y ) t ( F x , where t is a parameter, and if , b x when β t ; a x when α t ) t ( ' F dt dx is a continuous function on ] β , α [ , and ) t ( ' F does not change sign is in ) β , α ( , then the area of the region bounded by the curve ) t ( G y ) t ( F x , the x-axis and the lines a x , b x is dt dt ) t ( ' x y ) t ( ' F ) t ( G β α β α . ( Integration by substitution ) This formula is also true when if β α . In this case 0 ) t ( ' F dt dx

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Application of Definite Integral - Applications of Definite...

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