16.2

3)
F=<t^-2,t^2> dr=<1,-1*t^-2>
<t^-2,t^2>.<1,-t^-2>=t^-2+-t^0=t^-2-1
int(t^-2-1,t=1..2)=-t^-1-t,t=1..2=(-1/2+1)-1=-1/2

9)
ds=sqrt(1^2+(3x^2)^2)
int(sqrt(1+9*x^4)*sqrt(1+9x^4),t=0..1)
=14/5

11)
dr=<2,3,4>
int(16*t^2*sqrt(4^2+3^2+2^2))
=229.76

13)??
AB=0,0,1+t(0,2,-1)=0,2t,1-t -> 0*(0,2,-1) int(0*sqrt(2^2+1^2))=0
BC=0,2,0+t(1,-1,1)=t,2-t,t -> t*exp(t^2) int(t*exp(t^2)*sqrt(1^2+1^2+1^2))=1.48807
CA=1,1,1+t(-1,-1,0)=1-t,1+t,1 ->1-t*exp(1) int((1-t)*exp(1)*sqrt(1^2+1^2))=1.92

17)
represents length of curve distance between points
ds=sqrt(4^2+-3^2+12^2),t=2..5
=39

27)
ds=sqrt(1^2*2*x^2)
int(x^2*1-x*2*x,x=0..2)=-8/3

29)
0,0,0+t(1,4,4)=t,4t,4t
int((t-4*t)*1+(4*t-4*t)*4+4*t*4,t=0..1)
=13/2

31)??
1,0+t(0,1)=1,t
int((-t*0+1*1)/(1+t^2),t=0..1)
=pi/4


35) IDK


16.3

1)
F=<ysin(yz),xsin(yz)+xycos(yz)*z,xycos(yz)*y>
1*sin(pi)-0
=0

3)
3,6y
dr=1,-2t^-2
<3,6y>-> <3,12*t^-1>.<1,-2*t^-2>=3+-24*t^-3
int(3+-24*t^-3,t=1..4)
=-9/4

5)
ye^z,xe^z,xye^z
ye^z,xe^z,xye^z . 2t,3t^2,1= 2t^4*exp(t-1)+3t^4*exp(t-1)+t^5*exp(t-1)
=32e-1

9)
2y=y e^z=e^z 0=0
int(y^2,dx)=y^2*x+g(z,y)
2yx+g'(y)=2xy+e^z so g'(y)=e^z so g(y)=e^z*y ->y^2*x+y*e^z+g(z)
y*e^z+g'(z)=y*e^z so g'(z)=0 -> y^2*x+y*e^z

13)
0=0 1=1 sec^2(x)=sec^2(x)
int(zsec^2(x),dx)

15)
2x=2x -4=-4 0=0
int(2xy+5,dx)=yx^2+5x+g(y,z)
x^2+g'y=x^2-4z so g'y=-4z so g(y)=-4zy -> yx^2+5x-4zy+g(z)
-4y+g'z=-4y g'z=0 ->yx^2+5x-4zy

17)??
int(2*t^2*sin((pi*t)/4)*exp(t^2-2*t)*2*t+t^4*exp(t^2-2*t)*cos((pi*t)/4)*(pi/4)+t^4*sin((pi*t)/4)*exp(t^2-2t)*2*t-2,t=0..2)

19)
F= 2xy,x^2,-1= 2t^2,t^2,-1 . 1,1,0 =2t^2+t^2
int(2t^2+t^2,t=0..1)