Lecture13_Double_integrals_in_polar_coor

Lecture13_Double_integrals_in_polar_coor - Double Integrals...

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6 Double Integrals in Polar Coordinates ± The integral over the region R may be calculated by 2 successive integrations using two different orders: ± The element of area in polar coordinates is given by 2 2 2 ); sin( ); cos( y x r r y r x d dr r dy dx dA + = = = = = θ with ∫∫ ∫∫ ∫ ∫ ∫∫ = = = b a r h r h R g g R rdr d r u d r rd r u rdrd r r f dxdy y x f β α ) ( ) ( ) ( ) ( 2 1 2 1 ) , ( ) , ( ) sin , cos ( ) , ( Å r - f i s t
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7 Example 1 (P9-11.5) Find the volume bounded by; r = 5 cos 3 θ , z = 0 and z = 3 Use symmetry, i.e., integrate over half of the area 3D Projection in the xy-plane Sketch the projected area. Complete the solution
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8 Example 2 (P9-11.26) Find the integral by changing to polar coordinates Write in the integral polar coordinates Sketch the projected area. Complete the solution
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9 Example 3 (P9-11.34) Find the integral over R Since the region is circular, change to polar coordinates Complete the solution
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10 Example 4 (P9-11.17) Find the moment of inertia of the lamina bounded by r = a,
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Lecture13_Double_integrals_in_polar_coor - Double Integrals...

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