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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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