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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
= 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
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online