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Section 17.2 Line Integrals
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### Transcript of Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤...

• Section 17.2Line Integrals

• LINE INTEGRALSLet C be a smooth plane curve given byx = x(t), y = y(t), a t b.We divide the parameter interval [a, b] into n subintervals [ti 1, ti]of equal width, and we let xi = x(ti) and yi = y(ti). Then the corresponding points Pi(xi, yi) divide C into n subarcs with lengths si. Let be a point on the subarc Ci. If f is defined on a smooth curve C, then the line integral of f along C

if the limit exists.

• EVALUATING LINE INTEGRALSRecall from Section 11.2 that the arc length of C is

If f is a continuous function, the limit on the previous slide always exists. The following formula can be used to evaluate the line integral.

• INTERPRETATION OF THELINE INTEGRALIf z = f (x, y) 0 and C is a curve in the plane, then line integral

gives the area of the curved curtain below the surface and above C. See Figure 2 on page 1099.

• EXAMPLEEvaluate the following line integral where C is the line segment joining (1, 2) to (4, 7)

• PIECEWISE-SMOOTH CURVES AND LINE INTEGRALSIf C is a piecewise-smooth curve then C can be written as a finite union of smooth curves; that is,C = C1 U C2 . . . U CnThe line integral of f along C is defined as the sum of the line integrals of f along each of the smooth pieces of C; that is,

• EXAMPLEEvaluate where C is the piecewise- smooth curve formed by the boundary region bounded by y = x and y = x2.

• AN INTERPRETATION OF THE LINE INTEGRALSuppose that (x, y) represents the density of a thin wire that is shaped like the plane curve C. The mass of the wire is given by

The center of mass of the wire is given by

• EXAMPLEA thin wire is bent in the shape of the semicirclex = cos t, y = sin t, 0 t If the density of the wire at a point is proportional to its distance from the x-axis, find the mass and center of mass of the wire.

• LINE INTEGRALS WITH RESPECT TO x AND yTwo other line integrals can be obtained by replacing si by either xi = xi xi 1 or yi = yi yi 1. They are called the line integrals of f along C with respect to x and y.

• DISTINGUISHING FROM THE ORIGINAL LINE INTEGRALTo distinguish the line integral with respect to x and y from the original line integral C f (x, y) ds, we call C f (x, y) ds the line integral with respect to arc length.

• EVALUATING LINE INTEGRALS WITH RESPECT TO x AND y

• A SPECIAL NOTATIONThe line integrals with respect to x and y frequently occur together. We write this as follows.

• ORIENTATION AND LINE INTEGRALSRecall that a given parametrization x = x(t), y= y(t), a t b, determines an orientation of a curve C. If we let C denote the curve consisting of the same points as C but with opposite orientation, then we have:NOTE: The line integral with respect to arc length DOES NOT change sign.

• LINE INTEGRALS IN SPACESuppose that C is a smooth space curve given byx = x(t), y = y(t), z = z(t), a t b.Suppose that f is function of three variables that is continuous on some region containing C, then the line integral of f along C is defined in a similar manner as for plane curves:

• EVALUATING LINE INTEGRALS IN SPACE

• VECTOR NOTATION FOR LINE INTEGRALSIf r(t) is the vector form of either a plane curve or a space curve, then the formula for evaluating a line integral with respect to arc length can be written compactly as

• WORK AND LINE INTEGRALSSuppose that F = P i + Q j + R k is a continuous force field in three dimensions, such as a gravitational field. To compute the work done by this force in moving a particle along the smooth curve C, we divide C into subarcs Pi1Pi with lengths si by dividing the parameter interval [a, b] into subintervals of equal width. Choose a point on the ith subarc corresponding to the parameter . If si is small, then as the particle moves from Pi1 to Pi along the curve, it proceeds approximately in the direction of , the unit tangent vector at . The work done by the force F in moving to particle from Pi1 to Pi is approximately

The total work done in moving the particle along C is approximately

• WORK (CONCLUDED)Based on the derivation on the previous slide, we define the work W done by the force field F in moving a particle along C as the limit of the Riemann sums, namely,

• EVALUATING A LINE INTEGRAL FOR WORKIf the curve C is given by the vector equationr(t) = x(t)i + y(t)j + z(t)kthen T(t) = r(t)/|r(t)|. So, the line integral for work can be rewritten as

• EXAMPLEFind the work done by the force field

on a particle as it moves along the helix given byr(t) = cos ti + sin tj + tkfrom the point (1, 0, 0) to (1, 0, 3)

• LINE INTEGRAL OF A VECTOR FIELDDefinition: Let F be a continuous vector field defined on a smooth curve C given by a vector function r(t), a t b. Then the line integral of F along C is

where F = Pi + Qj + Rk.

• NOTE 1Even though C F dr = C F T ds and integrals with respect to arc length are unchanged when orientation is reversed, it is still true that

because the unit tangent vector T is replaced by its negative when C is replaced by C.

• NOTE 2where F = Pi + Qj + Rk