Calculate the integral of line $\displaystyle \int_c F.dr$, where $\displaystyle F(x,y) = \sqrt{x^2+y^2}$ and c is the circumference $\displaystyle x^2+y^2=1$
How do I calculate integral of line ?
Apprentice
In English we call it a line integral
The question only makes sense If F is a vector field
Do you mean F = r = xi +yj ?
In which case use x = cos(t) y = sin(t) to parameterize the curve
r = cos(t) i +sin(t) j dr/dt = -sin(t) i +cos(t) j
On the is curve F = cos(t) i +sin(t) j
F*dr/dt = 0
Therefor the line integral is 0
If you want check out the line integral page on my website for the general method
Line Integrals
dr is understood to be the vector dx i +dy j
Recall For line integrals to actually calculate we use
integral of F(x(t),y(t)) *dr/dt as t varies from a to b
F*dr is just notation rarely used just like the notation integral(fdx+gdy)
where F = f i + g j
We use integral of F(x(t),y(t)) *dr/dt
Again check out
Line Integrals for the development of line integrals
This problem i find in internet but not correct.
This problem is the book:
1) C is the curve represented by the equations:
$\displaystyle x=2t; y=3t^2; (0 \leq t \leq 1)$
Calculate the line integral along C
a) $\displaystyle \int_c (x-y)ds$
b) $\displaystyle \int_c (x-y)dx$
c) $\displaystyle \int_c (x-y)dy$