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Home Vector Calculus Line Integrals Theorem: Evaluating Line Integrals
 
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Theorem: Evaluating Line Integrals

The next theorem is useful for evaluating line integrals. It shows that any other parameter t can be used in place of the length s of the curve. Figure 13.2.6 illustrates the four parts of this theorem.

THEOREM

Let ∫C F · dS be a line integral.

(i)

If C is a horizontal directed line segment x0 ≤ x ≤ x1, y = y0, then

13_vector_calculus-67.gif

13_vector_calculus-63.gif

(i) Horizontal
(ii)

If C is a vertical directed line segment x = x0, y0 ≤ y ≤ y1, then

13_vector_calculus-68.gif

13_vector_calculus-64.gif

(ii) Vertical
(iii)

If C is traced by a parametric curve x = g(t), y = h(t), c0 ≤ t ≤ C1 where dx/dt and dy/dt are continuous, then

13_vector_calculus-69.gif

13_vector_calculus-65.gif

(iii) Parametric curve
(iv)

Reversing the curve direction changes the sign of a line integral. That is, if C1 is the curve C with its direction reversed, then

13_vector_calculus-70.gif

13_vector_calculus-66.gif

(iv) Reversing the curve direction Figure 13.2.6

Remark The integrals 13_vector_calculus-71.gifare sometimes called partial integrals.

PROOF

(i) and (ii) are special cases of (iii).

(iii) is proved by a change of variables,

13_vector_calculus-72.gif

(iv) is true because reversing the limits changes the sign of an ordinary integral.

Example 1: Horizontal and Vertical Line
Example 2: Work Done by a Force Vector Along a Curve

Home Vector Calculus Line Integrals Theorem: Evaluating Line Integrals

Last Update: 2006-11-20