15-4 - Math 200 Spring 2010 Handout#20 Section 15-4...

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Math 200, Spring 2010 Handout #20 Section 15-4 Integration in Polar, Cylinderical, and Spherical Coordinates Double Integrals in Polar Coordinates The polar coordinates ( ) , r θ of a point are related to the rectangular coordinates ( ) , x y by equations: 2 2 2 cos sin r x y x r y r θ θ = + = = . If the polar rectangle ( ) { } 1 2 1 2 , | , R r r r r θ θ θ θ = is divided into mn polar subrectangles ij R of equal widths ( ) 1 2 1 i i r r r r r m Δ = = and ( ) 1 2 1 j j n θ θ θ θ θ Δ = = , and centers ( ) * 1 1 2 i i i r r r = + and ( ) * 1 1 2 j j j θ θ θ = + , then the area of subrectangle ij R is 2 2 * 1 1 1 2 2 i i i i A r r r r θ θ θ Δ = Δ Δ = Δ Δ and a double Riemann sum over polar rectangle R is ( ) ( ) * * * * * 1 1 * * * * 1 1 cos sin cos sin , , i i m n i j i j i j m n i j i j i j r r f r r A f r r θ θ θ θ θ = = = = = Δ Δ Δ ∑∑ ∑∑ The double integral ( ) , R f x y dA ∫∫ is the limit ( ) ( ) 2 2 1 1 * , * * * * 1 1 lim cos sin cos , sin , r i r m n m n i j i j i j r r f r r rdrd f r r θ θ θ θ θ θ θ θ →∞ = = = Δ Δ ∫ ∫ ∑∑ 1. Evaluate the integral 4 2 2 6 rdrd π π θ . Sketch the region whose area is given by the integral.
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Changing to Polar Coordinates in a Double Integral If f is continuous on a polar rectangle ( ) { } 1 2 1 2 , | , R r r r r θ θ θ θ = then ( ) ( ) 2 2 1 1 , cos , sin r r R f x y dA f r r rdrd θ θ θ θ θ == ∫ ∫ ∫∫ To convert from rectangular to polar coordinates in a double integral, write cos x r θ = and sin y r θ = , use appropriate limits of integration for r and θ , and replace dA by rdrd θ . 2. Evaluate the integral ( ) R x y dA + ∫∫ where R is the region that lies to the left of the y -axis between the circles 2 2 1 x y + = and 2 2 4 x y + = 3. Use polar coordinates to find the volume of the solid that is below the paraboloid 2 2 18 2 2 z x y = and above the xy -plane.
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