lecture13

lecture13 - θ rdrd θ Lecture 13 Polar Coordinates Class...

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Lecture 13: Polar Coordinates May 29, 2009 Lecture 13: Polar Coordinates

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Objectives 1 Convert equations between polar and rectangular coordinates. 2 Calculate double integrals in polar coordinates. Lecture 13: Polar Coordinates
Polar Coordinates Change from rectangular to polar coordinates: x = r cos θ y = r sin θ Lecture 13: Polar Coordinates

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Polar Coordinates Change from rectangular to polar coordinates: x = r cos θ y = r sin θ Change from polar coordinates to rectangular coordinates r 2 = x 2 + y 2 tan θ = y x Lecture 13: Polar Coordinates
Double integrals in polar coordinates “Rectangles” are deﬁned by R = { ( r , θ ) | a r b , α θ β } . Lecture 13: Polar Coordinates

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Double integrals in polar coordinates “Rectangles” are deﬁned by R = { ( r , θ ) | a r b , α θ β } . dA = r drd θ Lecture 13: Polar Coordinates
Double integrals in polar coordinates “Rectangles” are deﬁned by R = { ( r , θ ) | a r b , α θ β } . dA = r drd θ ZZ R f ( x , y ) dA = Z β α Z b a f ( r cos θ, r sin

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Unformatted text preview: θ ) rdrd θ Lecture 13: Polar Coordinates Class exercise Find the volume of the solid bounded by the paraboloid z = 1 + 2 x 2 + 2 y 2 and the plane z = 7. Lecture 13: Polar Coordinates Class exercise Find the volume of the solid bounded by the paraboloid z = 1 + 2 x 2 + 2 y 2 and the plane z = 7. Answer: 9 π 4 Lecture 13: Polar Coordinates Using polar coordinates for general regions. Deﬁne a region as D = { ( r , θ ) | α ≤ θ ≤ β, h 1 ( θ ) ≤ r ≤ h 2 ( θ ) } , then ZZ D f ( x , y ) dA = Z β α Z h 2 ( θ ) h 1 ( θ ) f ( r cos θ, r sin θ ) rdrd θ Lecture 13: Polar Coordinates Class exercise Z a Z-√ a 2-y 2 x 2 ydxdy Lecture 13: Polar Coordinates Class exercise Z a Z-√ a 2-y 2 x 2 ydxdy Answer: a 5 15 Lecture 13: Polar Coordinates...
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This note was uploaded on 04/29/2010 for the course MATH 200 taught by Professor Unknown during the Spring '03 term at UBC.

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lecture13 - θ rdrd θ Lecture 13 Polar Coordinates Class...

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