Question

7. (5 pts) By completing the limits and integrand, set up (without evaluating) an iterated inte- gral which represents the volume of the ice cream cone bounded by the cone z = V x2 + y2 and the hemisphere z = V8 – x2 - y2 using (a) Cartesian coordinates. volume = dz dx dy. (b) polar coordinates. volume = I dr de.

Answer 1

2 = Juhty 2 -> Z = J 8-1?y? Q.7 By Completing the limits and integrand, set up (without evaluating) an iterated integral which Represents the volume cef the ice cream come bounded by the cone Z= √ x² + y² and the hemisphere z = √ 8-1²-y² using [a] Cartesian Coordinates Volume - w du Z У om So our z is to fro Ju²ty2 <reſ 8-4²_y2.

for limit of n and y first find the intersection of those two surfaces and then taking projection of this intersection onto Xy plane M²+y2 58-n²y2 n2+ y2 = 8-12_y2 2(x² + y2) = 8 =) (-2,0) -D (2,0) n²+ y²=4 Which is a circle. Now taking projection of this circle on xy plane. for limit of you ni -54.y? <ne S4-y? and yo - 2 ye2 2 (4-42 hence volume is 2 Volume 8-12 y 2 dZ dndy -2 -54-92 & 12 + y2 [1.] 1

[6.] Polar Coordinates Amy : By ean [1. 54 92 / 8 - 12 y z Volume dz dyndy 2 2 - 54-y2J n² ty? 2 54-y? 8-12-12 -2. Now Chang it into Polar coordinates. 71= cose, yo Simo dudy rardo a) ean of circle n²+ y2 = 4 0 e = 2 force covering this circle and oc rc2 oso<2T our volume is 2T 2 8- sos'o-t? Sino - Se cose + ° Sempo rdedo

"JATT.) e²] dedo drdo Ang