Chapter 9: Vector Calculus
Section 9.5: Line Integrals
Obtain the line integral of the scalar function fx,y=x y, taken around the circle whose center is 3,5 and whose radius is 2.
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∫02 πfR dt = 2∫02 π3+2 cost 5+2 sint dt = 60 π
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 9.5.5(a).
Context Panel: Evaluate Integral
x y→line integral∫02⁢π2⁢3+2⁢cos⁡t⁢5+2⁢sin⁡t⁢sin⁡t2+cos⁡t2ⅆt=60⁢π
Figure 9.5.5(a) Path Integral Domain dialog
Form and evaluate the line integral via the PathInt command
PathIntx y,x,y=Circle3,5,2 = 60⁢π
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 ∫02 πfR ρ dt
Calculus palette: Definite integral operator
2∫02 πfR ⅆt = 60⁢π
Explicit formulation and evaluation of the line integral
Write the integrand and press the Enter key.
Context Panel: Constructions≻Definite Integral≻t
Context Panel: Evaluate (Integral)
→integrate w.r.t. t
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