10.27 - Return HW Section 12.3: Double integrals in polar...

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Return HW Section 12.3: Double integrals in polar coordinates Main ideas? ..?. . Polar coordinates The definition of a polar rectangle The area of a “differential polar rectangle” Integration of functions over polar rectangles: ∫∫ A f ( x , y ) dx dy ∫∫ P f ( r cos θ , r sin ) r dr d Integration of functions over general polar regions Some integrals are easier to compute in polar coordinates Problem: Find the volume of the solid that lies under the paraboloid z = ( x +1) 2 + y 2 , above the xy -plane, and inside the cylinder x 2 + y 2 = 1. The solid lies above the disk D of radius 1 centered on the origin: a polar rectangle {( r , ): 0 2 π , 0 r 1}. We put x = r cos and y = r sin , and write ∫∫ D ( x +1) 2 + y 2 dA = 0 2 0 1 [( r cos + 1) 2 + ( r sin ) 2 ] r dr d
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= 0 2 π 0 1 [ r 2 cos 2 θ + 2 r cos + 1 + r 2 sin 2 ] r dr d = 0 2 0 1 ( r 2 + 2 r cos + 1) r dr d = 0 2 0 1 ( r 3 + 2 r 2 cos + r ) dr d = 0 2 0 1 (2 r 2 cos ) dr d + 0 2 0 1 ( r 3 + r ) dr d = ( 0 1 2 r 2 dr ) ( 0 2 cos ) + ( 0 1 r 3 + r dr ) ( 0 2 1 d ) = ((2 r 3 /3 | 0 1 ) (sin | 0 2 ) + ( r 4 /4 + r 2 /2 | 0 1 ) 2 = (2/3) (0) + (1/4 + 1/2) 2
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10.27 - Return HW Section 12.3: Double integrals in polar...

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