On these types of problems only, for answers that are numerically correct but do

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On these types of problems only, for answers that are numerically correct but do not have the correct number of significant digits. If a problem requires an exact answer, an approximated (rounded) answer will not be accepted. The MathPad or CalcPad tool that pops up will assist you on questions that require a correctly formatted mathematical expression (ie. expressions involving squareroots, Euler's number, greek symbols, trig functions, etc.) Note that variables (letters and symbols) are case sensitive. The message. "Your answer cannot be understood or graded" will appear when a syntax error (ie, missing operands, incorrect grouping operators, misspelled units, symbols with no meaning in response) has been made. You will not lose submissions for syntax errors. Please see the topic. Answering Math and Science Questions, under the WebAssign Student Help System for detailed information and examples. Let F(x, y, z) = z tan^-1 (y^2)i + z^3 ln(x^2 + 3)j + zk. Find the flux of F across S, the part of the paraboloid x^2 + y^2 + z = 14 that lies above the plane z = 5 and is oriented upward. integral integral_S F middot dS =

Explanation / Answer

divF=0+0+1 =1

divF=1

x2+y2+z=14 ,z=5

x2+y2+5=14

x2+y2=9

x2+y2=32

x2+y2+z=14 ,z=5

z=14-x2-y2 ,z=5

in cylindrical coordinates

x=rcos y=rsin

x2+y2=r2

0<=<=2,0<=r<=3 ,5<=z<=14-r2

dv =r dz dr d

by divergence theorem sF.dS=EdivF dv

[0 to 2] [0 to 3] [5 to 14-r2] 1r dz dr d

[0 to 2] [0 to 3][5 to 14-r2] r z dr d

[0 to 2] [0 to 3]  r(14-r2-5) dr d

[0 to 2] [0 to 3] (9r-r3) dr d

[0 to 2][0 to 3] ((9/2)r2-(1/4)r4) d

[0 to 2] ((9/2)32-(1/4)34)-0 d

[0 to 2] (81/4)

=(81/4)(2-0)

=(81/2)

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