CALCULUS Understanding Its Concepts and Methods

Line integral

Let
*C*
be the curve
(*x*(*t*),*y*(*t*)),
*a ≤ t ≤ b*,
and *f*(*x*,*y*)
a scalar field such that
*f*(*x*(*t*),*y*(*t*)) ≥ 0
for
*a ≤ t ≤ b.*

The line integral
is
the surface area of a curtain whose height is
*f*(*x*(*t*),*y*(*t*))
above the point
(*x*(*t*),*y*(*t*)).

The line integral can be evaluated by converting the it into an ordinary integral using

Let
*C*
be the space curve
(*x*(*t*),*y*(*t*),*z*(*t*)),
*a ≤ t ≤ b*,
and let
*f*(*x*,*y*,*z*)
be a scalar field.

The line integral is given by

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.