Question

I need a help on how to evaluate line integral,eg
Evaluate the line integral x^3dx +3y^2dy-x^2ydz
from P(3,2,1) to Q(6,4,2) using parametric representations.
My problem is that How to get those parameters?

Answer 1

\[ x^3dx +3y^2dy-x^2ydz = (x^3 \vec i + 3y^2 \vec j + x^2y\vec j).(\vec idx + \vec jdy + \vec k dz)\]

Answer 2

http://www.wolframalpha.com/input/?i=curl+ [x^3%2C++3y^2%2C+-x^2y]
This is path dependent ... I think you need a path ...

Answer 3

You mean like here?
http://answers.yahoo.com/question/index?qid=20100404183015AAezoDM

Answer 4

haha ... thats a path in straight line!!

Answer 5

To parametrize a line from a point p1 to a point p2 you write
r(t) = p1 + t (p2-p1) where t is between 0 and 1.
In our case
\[
r(t)= P+ t(Q-P)=\{3 t+3,2 t+2,t+1\}
\]

Answer 6

So
{x(t),y(t),z(t)}={3t+3,2t+2,t+1}
Now\[
F(x(t),y(t),z(t))=\left\{(3 t+3)^3,3 (2
t+2)^2,(-2 t-2) (3
t+3)^2\right\}=\\
\left\{27 t^3+81 t^2+81
t+27,12 t^2+24 t+12,-18
t^3-54 t^2-54 t-18\right\}\\
F(x(t),y(t),z(t)) .\{x'(t),y'(t),z'(t)\}=\\
63 t^3+213 t^2+237 t+87
\]
Integrate the above from t=0 to t=1. You will be done

Answer 7

thanx eliassaab, but make it more clear to me, is it a must that t should be in [[0,1]?

Answer 8

This particular choice requires t to be between 0 and 1.