Chapter 9: Vector Calculus
Section 9.5: Line Integrals
Let C be the circle whose center is 3,5 and whose radius is 2. Let c be the arc on C subtended by a central angle of π/4 radians measured counterclockwise from the right half of a horizontal diagonal. Obtain the line integral of the scalar function fx,y=x y, taken along c.
A parametric definition of the circle as a position vector is given by
R=3+2 cos(t)5+2 sin(t)
For this circle, ρ=dsdt=R. is given by
ddt3+2 cost2+ddt5+2 sint2
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 around the circle is ρ fR. Hence, the line integral is
2∫0π/4fR dt = 2∫0π/43+2 cost 5+2 sint dt = 14+152⁢π+4⁢2 ≐ 43.22
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 9.5.6(a).
Context Panel: Evaluate Integral
x y→line integral∫014⁢π2⁢3+2⁢cos⁡t⁢5+2⁢sin⁡t⁢sin⁡t2+cos⁡t2ⅆt=14+152⁢π+4⁢2
Figure 9.5.6(a) Path Integral Domain dialog
Form and evaluate the line integral via the PathInt command
PathIntx y,x,y=ArcCircle3,5,2,0,π/4 = 14+152⁢π+4⁢2
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 circle parametrically as the position vector R
Context Panel: Assign Name
R=3+2 cost,5+2 sint→assign
Calculus palette: Differentiation operator
Context Panel: Evaluate and Display Inline
ⅆⅆ t R = 2
Form and evaluate the line integral ∫0π/4fR ρ dt
Calculus palette: Definite integral operator
2∫0π/4fR ⅆt = 14+152⁢π+4⁢2
Explicit formulation and evaluation of the line integral
Write the integrand and press the Enter key.
Context Panel: Constructions≻Definite Integral≻t
Context Panel: Approximate≻10 (digits)
→integrate w.r.t. t
→at 10 digits
<< 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)