Using the Fundamental Theorem of Line Integrals In Exercises 25–34, evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ∫ C e x sin y d x + e x cos y d y C : cycloid x = θ − sin θ , y = 1 − cos θ from ( 0 , 0 ) to ( 2 π ,0 )
Using the Fundamental Theorem of Line Integrals In Exercises 25–34, evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.
∫
C
e
x
sin
y
d
x
+
e
x
cos
y
d
y
C
:
cycloid
x
=
θ
−
sin
θ
,
y
=
1
−
cos
θ
from
(
0
,
0
)
to
(
2
π
,0
)
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
A. State the F undamental Theorem of Calculus for Line Integrals.
B. Let f(x, y, z) = xy + 2yz + 3zx and F = grad f. Find the line integral of F along the line C with parametric equations
x = t, y = t, z = 3t, 0 ≤ t ≤ 1.
You must compute the line integral directly by using the given parametrization.
C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.
Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals.
φ(x, y) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2
Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals.
φ(x, y, z) = (x2 + y2 + z2)/2; C: r(t) = ⟨cos t, sin t, t/π⟩ , for 0 ≤ t ≤ 2π
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