Chapter 9: Vector Calculus
Section 9.5: Line Integrals
Let C be the ellipse x2+4 y2=1 and let c be the arc subtended by an angle of π/4 radians measured counterclockwise from the positive x-axis. Obtain the line integral of the scalar function fx,y=x y, taken along c.
A parametric definition of the ellipse as a position vector is given by
R= cos(t) sin(t)/2
since x2+2 y2=cos2t+2 sin2t/2=1.
For this ellipse, ρ=dsdt=R. is given by
ddt cost2+ddt sint/22
=4 sin2t+2 cos2t/2
If fv=v1⋅v2 is the scalar-valued function of the vector argument v=v1 i+v2 j, then the integrand for the line integral along the ellipse is ρ fR. Before the requisite line integral can be formulated, however, the value of t that subtends a "polar angle" of π/4 radians must be determined. Where the ray y=x hits the ellipse in the first quadrant, cost=sint/2, so that t=arctan2. Hence, the line integral is
∫0arctan2ρ fR dt = 122∫0arctan2 costsint 4−2 cos2t dt = 151554−16
Tools≻Load Package: Student Vector Calculus
Access the PathInt command through the Context Panel
Write the scalar.
Context Panel: Student Vector Calculus≻Line Integral (2D)
Complete the dialog as per Figure 9.5.8(a).
Context Panel: Evaluate Integral
x y→line integral∫0arctan⁡214⁢cos⁡t⁢sin⁡t⁢4⁢sin⁡t2+cos⁡t2ⅆt=−136+17900⁢85
Figure 9.5.8(a) Path Integral Domain dialog
Form and evaluate the line integral via the PathInt command
PathIntx y,x,y=ArcEllipsex2+4 y2−1,0,π/4,output=integral
PathIntx y,x,y=ArcEllipsex2+4 y2−1,0, π/4 = −136+17900⁢85
A solution from first principles is also possible.
Define f as a scalar-valued function of a vector argument
Context Panel: Assign Function
fv=v1⋅v2→assign as functionf
Define the ellipse parametrically as the position vector R
Context Panel: Assign Name
R= cost, sint/2→assign
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻rho
ⅆⅆ t R = 12⁢−3⁢cos⁡t2+4→assign to a nameρ
Form and evaluate the line integral ∫0arctan2fR ρ dt
Calculus palette: Definite integral operator
∫0arctan2ρ fR ⅆt = −136+17900⁢85
Explicit formulation and evaluation of the line integral
Write the integrand and press the Enter key.
Context Panel: Constructions≻Definite Integral≻t
(Complete dialog as per figure to the right.)
→integrate w.r.t. t
<< Previous Example Section 9.5
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)