Calculus help

[junhoma]

I have to evaluate integral C (y/x) ds where C is along the line y = 2x from points ( 0,0 ) t0 ( 2, 4).

I did like this

x= 2t and y = 4t so t goes from o to 1
and i got answer 4sqrt 5 is it right answer?. did i do right parametric equation please.

 

[galactus]

Looks OK.

You can also do this without parameterizing.

Since y=2x, sub into y/x

\(\displaystyle \int_{C}\frac{2x}{x}=\int_{C}2ds\)

\(\displaystyle ds=\sqrt{(x')^{2}+([2x]')^{2}}=\sqrt{1+2^{2}}=\sqrt{5}\)

\(\displaystyle \int_{0}^{2}2\sqrt{5}dx=4\sqrt{5}\)

 

[junhoma]

Thank you somuch.