we are given
[tex]r(t)=(3t,3cost,3sint)[/tex]
we can compare it with
[tex]r(t)=(x,y,z)[/tex]
we get
[tex]x=3t,y=3cost,z=3sint[/tex]
now, we can use length formula
[tex]L=\int\limits^a_b {\sqrt{(x')^2+(y')^2+(z')^2} } \, dt[/tex]
now, we can find x' , y' and z'
[tex]x'=3,y'=-3sint,z'=3cost[/tex]
now, we can plug values
and we get
[tex]L=\int\limits^-4_4 {\sqrt{(3)^2+(-3sint)^2+(3cost)^2} } \, dt[/tex]
now, we can simplify it
[tex]L=\int\limits^-4_4 {\sqrt{(3)^2+9} } \, dt[/tex]
[tex]L=\int\limits^-4_4 {3\sqrt{2} } \, dt[/tex]
we get
[tex]L=24\sqrt{2}[/tex]............Answer
