Line integral question, answer is here, just confused on it

[mr_coffee]

Hello everyone I'm confused on this line integral.
The substiution is easy but I'm not sure where 2t is coming from...

integral over C x^2*y*sqrt(z) dz;
C: x = t^3;
y = t;
z = t^2;

0 <= t <= 1

integral over C x^2*y*sqrt(z) dz =
integral 0 to 1 (t^3)^2 (t) sqrt(t) * 2t dt =
integral 0 to 1 2*t^9 dt;

Okay I see they are just plugging in the t's for the x,y,z, but why do they write sqrt(t)? the sqrt(t^2) is just t.

Also where is this 2t dt coming from?

I thought maybe they used: sqrt(t^2); u = t^2;
du = 2t dt;
1/2*du = t dt;


Thanks

 

[neutrino]

I guess sqrt(t) is just a typo. It should be sqrt(t^2) (Note that the final integral is t^9). And that 2tdt is equal to the differential dz.

 

[arunma]

What Neutrino said. The final integrand should be 2t^9. In fact, if you compare step 2 to step 3, you'll find that it does not logically follow. So in all likelihood it was a typo.

 

[mr_coffee]

THanks for the help guys!