Chapter 9: Vector Calculus
Section 9.5: Line Integrals
Obtain the line integral of the scalar function fx,y=x y, taken along the polygonal line connecting the points 1,2, 3,5, 4,0, 2,7, in that order.
The line integral of the scalar fx,y along a path described parametrically by x=xt,y=yt, t∈a,b, is given by
∫abfxt,yt dsdt dt
where s is arc length, so dsdt=x.2+y.2=ρ = R., with Rt=xt i+yt j being the vector form of the parametric representation of the path. However, there are three linear segments for the polygonal line.
A parametric representation of the first line segment is
R=x(t)y(t)=12+t (35−12) = 1+2 t2+3 t,0≤t≤1
ρ=dsdt=ddt1+2 t2+ddt2+3 t2 = 22+32=13
and the line integral along this segment is given by
∫011+2 t 2+3 t 13 dt=13 ∫016⁢t2+7⁢t+2 dt=1513/2
A parametric representation of the second line segment is
R=x(t)y(t)=35+t (40−35) = 3+ t5−5 t,0≤t≤1
ρ=dsdt=ddt3+t2+ddt5−5 t2 = 12+52=26
∫013+t 5−5 t 26 dt=26 ∫0151−2 t−3 t2 dt=2526/3
A parametric representation of the third line segment is
R=x(t)y(t)=40+t (27−40) = 4−2 t7 t,0≤t≤1
ρ=dsdt=ddt4−2 t2+ddt7 t2 = 22+72=53
∫014−2 t 7 t 53 dt=53 ∫0114 2 t−t2 dt=2853/3
Thus, the line integral along the polygonal path is the sum
Tools≻Load Package: Student Vector Calculus
Access the PathInt command through the Context Panel
Write the scalar and press the Enter key.
Context Panel: Student Vector Calculus≻Line Integral (2D)
Complete the dialog as per Figure 9.5.3(a).
Context Panel: Evaluate Integral
Figure 9.5.3(a) Path Integral Domain dialog
Form and evaluate the line integral via the PathInt command
Clearly, integrating along a polygonal line is equivalent to integrating along each segment of the path, and adding the resulting integrals.
PathIntx y,x,y=Line1,2,3,5 = 152⁢13
PathIntx y,x,y=Line3,5,4,0 = 253⁢26
PathIntx y,x,y=Line4,0,2,7 = 283⁢53
A solution from first principles is also possible, but the bulk of the work consists in obtaining parametric representations of each line segment.
Apply the BasisFormat command.
Let P be a list of the four nodes given as vectors.
Obtain parametric representations of each line segment, and calculate ρ for each segment
Write the equation of a line segment and press the Enter key.
Context Panel: Student Vector Calculus≻Differentiate≻With Respect To≻t
Context Panel: Student Vector Calculus≻Norm≻Euclidean
Now write and evaluate three integrals of the form ∫01fxt,yt ρ dt.
∫011+2 t 2+3 t 13 ⅆt = 152⁢13
∫013+t 5−5 t 26 ⅆt = 253⁢26
∫014−2 t 7 t 53 ⅆt = 283⁢53
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