Double integrals

Question

Find the line integral with respect to arc length ∫C (9x+5y)ds, where C is the line segment in the xy-plane with endpoints P=(3,0) and Q=(0,2). Find a vector parametric equation r(t) for the line segment C so that points P and Q correspond to t=0 and t=1, respectively. r (t)=?

Answer 1

The easiest approach to to start with the second part.

The vector from P towards Q is (0-3 , 2 -0) = ( -3,2)

Thus a parametric expression for the line segment from P towards Q is

(r_x , r_y) = t ( -3,2) + (3,0) 0 < t <1

So x = 3 - 3 t and y = 2 t and 9 x + 5y = 27 -17 t

So the line integral is ∫ ds [ 27 - 17 t ] limits: 0 to 1

Since the contour C is a straight line segment, ds = dt sqrt(3² + 2²) = dt sqrt(13)

The line integral is thus sqrt(13) ( 27 - (1/2) 17 ) ~ 66.7

Answer 2

Whenever you have a line segment starting at (a,b) and ending at (c,d), the parametrization that's easiest to derive is r(t)=<a,b>+t<c-a,d-b>, 0≤t≤1. Then, whenever you take a line integral ds, that means that you replace the ds with |r'(t)|dt (that is, the length of the derivative/tangent vector for the parametrization r(t) for the curve you're integrating over), any instances of x with a+t(c-a), and any instances of y with b+t(d-b). Put all this information together and you should have an integral you can hand to a student just finishing first semester calculus.