# Chapter 8 - Chapter Chapter 8 Multiple Integrals Overview Â...

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Unformatted text preview: Chapter Chapter 8 Multiple Integrals Overview Â¡ Double Integral Â¡ Double Integrals Â¢ Properties of Double Integrals Â¡ Evaluation Â¢ Rectangular Region Â¢ Rectangular Regions Â¢ Type A region Â¢ Type B region Â¢ Type B region Overview Â¡ Double Integrals in Polar Coordinate Â¡ Double Integrals in Polar Coordinates Â¢ Circle Â¢ Ring Â¢ Ring Â¢ Sector of a Circle Â¢ Polar Rectangular Â¢ Polar Rectangular Â¢ Change of Variables Overview Â¡ Application of Double Integral Â¡ Application of Double Integrals Â¢ Volume Â¢ Surface Area Â¢ Surface Area Â¢ Mass and Center of Gravity Â¡ Triple Integral Â¢ Physical Meaning Â¢ Physical Meaning Â¢ Rectangular Region Double Integrals Integrals y = f ( x ) y x Area under curve b a a b Â¡ dx x f A b a Â³ ) ( Riemann Integrals ( ) y f x 2 ( ) f c 1 ( ) f c x a x 1 c 2 c 1 n c Â¡ n c 1 x 2 x 2 n x Â¡ 1 n x Â¡ n x b Divide [ , ] into equal intervals a b n n a b x Â¡ ' interval each of Length n 1 2 Area of rectangles ( ) ( ) ( ) n f c x f c x f c x ' Â¢ ' Â¢ Â¢ ' Â¡ Riemann Integrals ( ) y f x 1 ( ) f c x b The under the curve of ( ) from to y f x a b area a x 1 c 2 c 1 n c Â¡ n c 1 x 2 x 2 n x Â¡ 1 n x Â¡ n x b 1 ( ) n k k f c x | ' Â¦ of on [ , ] f a b Riemann sum Riemann Integrals ( ) y f x 1 ( ) f c x a x 1 c 2 c 1 n c Â¡ n c 1 x 2 x 2 n x Â¡ 1 n x Â¡ n x b a b Â¡ When , we have n o f Area of rectangles Area under the curve ( ) from to f x x a x b o n x ' interval each of Length Area of rectangles Area under the curve ( ) from to . f x x a x b o Riemann Integrals ( ) y f x 1 ( ) f c x a x 1 c 2 c 1 n c Â¡ n c 1 x 2 x 2 n x Â¡ 1 n x Â¡ n x b Let n o f x c f n k k n ' Â¦ of ) ( lim 1 The exact area A is given by 1 ( ) lim ( ) n b k a n k A f x dx f c x of ' Â¦ Â³ Double Integrals Â¦ Â³Â³ of ' n i i i i n R A y x f dA y x f 1 ) , ( lim ) , ( R ) , ( i i y x i A ' Note: Â³Â³ Note: double integral sign means we are integrating R Â³Â³ over a two-dimensional region. Geometrical Meaning d f b dA y x f Â³ ) , ( ) ( t x f ) , ( t y x f dx x f a Â³ ) ( R Â³Â³ y ) , ( y x f z z ) ( x f y y a b x R a d area under the curve volume under the surface x b c area under the curve over the interval [ a, b ] volume under the surface over the region R Double Integrals (Geometrical meaning) If ( ) 0 for all points ( ) in the definite integra f x y x y R t If ( , ) for all points ( , ) in , the definite integral ( , ) is equal to the volume under the surface R f x y x y R f x y dA t Â³Â³ ( , ) and above the plane over the region . z f x y x y R Â¡ z z = f ( x,y ) y d a c x b Double Integrals d y R ij y j-1 y j ) , ( * * ij ij y x c y 1 x a b x 1 x 2 x i-1 x i Double Integrals z )) , ( , , ( y x f y x d ) , ( * * i i y x f a y d c ij ij b x R ij Double Integrals z y x Â¦ Â³ ' n A f dA * * lim Â¦ Â³Â³ of ' i i i i n R A y x f dA y x f 1 ) , ( lim ) , ( Properties of Double Integrals dA y x g dA y x f dA y x g y x f R R R Â³Â³ Â³Â³ Â³Â³ Â¡ Â¡ ) , ( ) , ( )] , ( ) , ( [ ( , ) ( , ) , where is a const ant....
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Chapter 8 - Chapter Chapter 8 Multiple Integrals Overview Â...

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