Question

1. Using polar coordinates in the x-y plane, find the volume of the solid above the cone z r and below the hemisphere z= v8-r2. As a check the answer is approximately 13.88 but of course you have to calculate the exact answer 2. At the right is the graph of the 8-leafed rose r 1+2cos(40) Calculate the area of the small leaf. As a check the answer is 0.136 to 3 places of decimal (But of course you have to calculate the exact answer.) [Tech help: cos2(kO)d0 = 0+sin( 2k0)+ C ] 2 4k 3. Find the mass and centre of mass of the solid S inside the paraboloid z = x2 +y and beneath the plane z ifthe density is p(x, y, z) = z. Use two different approaches, first with cylindrical coordinates and second slicing the solid horizontally

Answer 1

Given Solf the Cone Z Y hemisphuru 8 - Y fial Vahue 3.8 8 the aad alro iven have pave Now we 2 8-Y 2 7 Tdr de 2. J2 ddy R,0) osO2TT 2 2 Y Put dt -dt dy

8, dt 2 2 t do 2 3 do 1 3 3 13,88 32 Ti 3

and have find the mars Sol nthis Solid S Cent mas (Pat a bolaid (plane) 2 + Given j Her se

M P,)dA dA 2 2 Y 2 -D Solid Mass Mx Now 3 V 3

+ 24 Noo My dy Y dy 2 J. 24 2 Ma Centu Now

My M. /2 S 8 8 CoOndinates Y Y Cos o Similanh formmulae and imit Sam lindwical Coatee and mal to frd mas