Simple Line Integral becomes troublesome

[gipc]

I have the parametrization of C
x=9cos(t)
y=9sin(t)
z=-8t

0<=t<=10*pi

and I have to calculate $$\int_c$$ 7x^2+4y^2 -5xy

after I transform this I get
(7*81cos^2(t) +4*81sin^2(t) -5*81*cos(t)*sin(t))*sqrt(-9sin^2(t)+9cos^2(t)-8) dt from 0 to 10*pi

Now that is a monster.
What did I do wrong?

 

[Reshma]

Obtain the values of x, y, z in terms of the boundary conditions for t, i.e. t = 0 and t = $10\pi$. It should be easier to solve.

 

[HallsofIvy]

When t= 0 x= 9, y= 0, z= 0.

When $t= 10\pi$, x= 9, y= 0, $z= -80\pi$.

I don't see how that helps at all!

gipc, you statement $\int_c 7x^2+ 4y^2- 5z^2$ is missing the differential. Is that with respect to dx, dy, dz, or the arclength ds, or what?

Because of the square root, I assume it is ds but it would not be what you give. it would be
$$\sqrt{81 sin^2(t)+ 81 cos^2(t)+ 64}dt= \sqrt{81+ 64}dt= \sqrt{145}dt$$

Did you forget that $sin^2(t)+ cos^2(t)= 1$?

 

[gipc]

yes, i did forget :)
thanks for that, that was the mistake that was holding me back