Problem 15  (section 19.2)
INTEGRATION help

Clear[fun,phi,theta,f,x,y,z];
fun[phi_,theta_]=(81*Sin[phi]^3*Cos[phi]*Cos[theta]^2+27*Sin[phi]^2*Cos[phi]*Sin[theta])

Integrate[fun[phi,theta],phi]

Integrate[fun[phi,theta],{phi,0,Pi/2}]

Integrate[fun[phi,theta],{theta,0,2*Pi},{phi,0,Pi/2}]

Problem 15  (section 19.2)
SUBSTITUTION help

Clear[x,y,z,phi,theta];
x=3*Sin[phi]*Cos[theta];
y=3*Sin[phi]*Sin[theta];
z=3*Cos[phi];
vectorF={x*z,0,y}

normalvector={Sin[phi]*Cos[theta],Sin[phi]*Sin[theta],Cos[phi]}

(vectorF.normalvector)*3^2*Sin[phi]

Expand[%]

Problem 50  (section 19.2) hints

The fifth problem on the in-class worksheet is almost the same as problem 50 in the homework assignment.  It just has "3" for "a", and substituted u=7.  See the computation below.

++++++++++++++++ Worksheet integrals +++++++++++++
Note: problems 1 and 4 have "F·n" constant, so you don't need to evaluate an integral.

2.

Clear[x,y,z]
Integrate[z,{y,0,1}];
Integrate[ Integrate[z,{y,0,1}] ,{z,-1,0}]

3.

Clear[x,y,z];
Integrate[Integrate[6*x,{x,-Sqrt[25-z^2],Sqrt[25-z^2]}],{z,-5,5}]
(* OR *)
Integrate[Integrate[6*Cos[theta]*r,{r,0,5}],{theta,0,2*Pi}]

5.

Integrate[Integrate[7*(1-r^2/9)*r,{r,0,3}],{theta,0,2*Pi}]
(* OR *)
Integrate[Integrate[7*(1-(y^2+z^2)/9),{y,-Sqrt[9-z^2],Sqrt[9-z^2]}],{z,-3,3}]