16.4 Ex 14 - 19

# 16.4 Ex 14 - 19 - 946 C H A P T E R 16 M U LTI P L E I N T...

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946 CHAPTER 16 MULTIPLE INTEGRATION (ET CHAPTER 15) 14. Z 3 0 Z 9 y 2 0 q x 2 + y 2 dx dy SOLUTION The region D is defned by the Following inequalities: 0 y 3 , 0 x q 9 y 2 y x 12 1 2 0 34 3 4 D We see that D is the quarter oF the circle x 2 + y 2 = 9, x 0, y 0. We describe D in polar coordinates by the Following inequalities: 0 θ π 2 , 0 r 3 The Function in polar coordinates is f ( x , y ) = p x 2 + y 2 = r . Using change oF variables we get Z 3 0 Z 9 y 2 0 q x 2 + y 2 = Z / 2 0 Z 3 0 r · rdr d = Z / 2 0 Z 3 0 r 2 dr d = Z / 2 0 r 3 3 ¯ ¯ ¯ ¯ 3 0 d = Z / 2 0 9 d = 9 · 2 = 4 . 5 15. Z 1 / 2 0 Z 1 x 2 3 x xdydx The region oF integration is described by the inequalities 0 x 1 2 , 3 x y p 1 x 2 D is the circular sector shown in the fgure. x D y 0 1 1 2 π 3 3 x y = 1 x 2 y = The ray y = 3 x in the frst quadrant has the polar equation r sin = 3 r cos tan = 3 = 3 ThereFore, D lies in the angular sector 3 2 . Also, the circle y = p 1 x 2 has the polar equation r = 1, hence D can be described by the inequalities 3 2 , 0 r 1 We use change oF variables to obtain Z 1 / 2 0 Z 1 x 2 3 x = Z / 2 / 3 Z 1 0 r ( cos ) rdrd = Z / 2 / 3 Z 1 0 r 2 cos = Z / 2 / 3 r 3 cos 3 ¯ ¯ ¯ ¯ 1 r = 0 d

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SECTION 16.4 Integration in Polar, Cylindrical, and Spherical Coordinates (ET Section 15.4) 947 = Z π / 2 / 3 cos θ 3 d = sin 3 ¯ ¯ ¯ ¯ / 2 / 3 = 1 3 ³ sin 2 sin 3 ´ = 1 3 Ã 1 3 2 !
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16.4 Ex 14 - 19 - 946 C H A P T E R 16 M U LTI P L E I N T...

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