Question

Can someone help me with this?

(1 point) The region W is the cone shown below. The angle at the vertex is 1/2, and the top is flat and at a height of 3. Write the limits of integration for Sw dV in the following coordinates (do not reduce the domain of integration by taking advantage of symmetry): (a) Cartesian: With a = b= C= d= e = and f= Volume = Sa So Sol d d d (b) Cylindrical: With a = b= C- d = and f = Volume = Sa Se Sot d d d (c) Spherical: With a b C= d= e and f = Volume = Sale Sale SA d d d

Answer 1

First note that is *3 Conversion from cortesian to cylindricalis (x, y, z) to (8,0, 3) Putrs Sorry tan o = 4/x => dandyda ride do do ms z * Conversion from carterian to spherical (2 y, z) to ce o, o Put is L = (x²+ y²+2 dzdody tan o => cas" / 2 / x2 + y2 + 22 ) =gsing do dodo CS Scanned with CamScanner

for spherical coordinates let x= psin & caso y = p sino sino = andy dz = gr sin odode dip za o cos where, s = dist from origin to the point P3. p=0 o = Yadar angle (as in polar coordinater); 0 50 52 a = angle from the zoones to say from origin through p. 3oEost so CS Scanned with CamScanner

X z' (a Cartesian is * an the ny plane is In the y z plane y=Z at kr/4 This angle st is 7/4 by symmetry = -2 Similarly, di xz planers & 9²2 z² = x = 1/2 Z Hence, - 2 y = z & the given top in So, ot z = 3 Hence, O b= 3 d: 2 - Z e - 2² - y² LE Velune SSS 1. 1. dr dy dz CS Scanned with CamScanner

2 =27 4* Я osz 23 So; 2² 8 - z & O= tail y It is clear from the projection on ny plane that Osrs z & o 20 sat a = bs 3 e- & Velene = sale ca fet derdo dz c. with © o is tove angle that the point a makes the hole (horitive Z-axis) So, 020 s 1/4 s is ditācookboest Reisin the origini o co= an o = g = 3 *sec d R! So, 30/8 casa a = iwe 3 Р c = e= 6=2T d = "1/4 f = 3x sec & => 0 = 3.48 f pe sin f do do do & d Volume = S C CS Scanned with CamScanner

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