16.4

7.
r=(2u+v)*i+(u-4v)*j+(3u)*k
Tu=2i+1j+3k=Tu(1,4)
Tv=1i-4j+0k=Tv(1,4)
Tu×Tv=
i	j	k
2	1	3
1	-4	0
=12i+3j-9k
N=<12,3,-9>
G(1,4)=(6,-15,3)
(12)(x-6)+3(y+15)-9(z-3)=0
12x+3y-9z=0
Ans.	Tu=2i+1j+3k	Tv=1i-4j+0k	N=<12,3,-9>	Tangent plane:12x+3y-9z=0

13.
ru=<cos(v), sin(v),1>
rv=<-usin(v),ucos(v),0>
ru×rv=<-ucos(v),-usin(v),u>
|ru×rv|=sqrt(2)*u
Int(Int(u((u*cos(v))^2+(u*sin(v))^2)*sqrt(2)*u,v=0..1),u=0..1
=Int(sqrt(2)*u^4,u=0..1)
=sqrt(2)/5
Ans.	sqrt(2)/5

15.
z^2+y=9
z=sqrt(9-y)
dS=sqrt(1+(-1/(2*sqrt(9-y))))=sqrt((4y-37)/y-9)/2dA
Int(Int(sqrt(9-y)(sqrt((4y-37)/y-9)/2),y=0..9),x=0..3)
=56.016
Ans.	56.016

19.
Int(e^(-z),z=0..4)
=1-e^(-4)
Int(Int((1-e^(-4))*r,r=0..4)t=0..2Pi)
=Int(2*(e^4-1)*e^(-4)),t=0..2Pi)
=4*(e^4-1)*e^(-4)*Pi
Ans.	4*(e^4-1)*e^(-4)*Pi

16.5

5.
P=y	Q=z	R=x
gx=-3	gy=4
Int(F*ds)=Int(Int(3y-4z+x),x=0..1),y=0..1
z=1-3x+4y
Int(F*ds)=Int(Int(3y-4(1-3x+4y)+x),y=0..1),x=0..1
=Int(13*x-21/2,x=0..1)
=-4
Ans.	-4

7.
z=sqrt(9-x^2-y^2)
P=0,	Q=3	R=x
gx=-x/sqrt(-x^2-y^2+9)	gy=-y/sqrt(-x^2-y^2+9)
Int(F*ds)=Int(Int(3y/sqrt(-x^2-y^2+9)+x,)y=0..sqrt(9-x^2))x=0..3
=Int(F*ds)=Int((x+3)*sqrt(9-x^2),x=0..3)
=27*Pi/4+9
Ans.	27*Pi/4+9

9.
z=9-x^2-y^2
P=z	Q=z	R=x
gx=-2x	gy=-2y
Int(F*ds)=Int(Int(2xz+2yz+x),y=0..sqrt(9-x^2),x=0..3)
=Int(Int(2x(9-x^2-y^2)+2y(9-x^2-y^2)+x),y=0..sqrt(9-x^2),x=0..3)
=Int((x^2-9)^2/2-x*(4*x^2-39)*sqrt(9-x^2)/3,x=0..3)
=693/5
Ans.	693/5

11.
z=1-x-y
P=y^2	Q=2	R=-x
gx=-1	gy=-1
Int(Int((y^2+2-x),y=0..1-x),x=0..1)
=Int(-(x-1)*(x^2-5*x+7)/3,x=0..1)
=11/12
Ans.	11/12