Line integrals Use Green’s Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 32. ∮ C 〈 sin y , x 〉 ⋅ d r , where C is the boundary of the triangle with vertices (0, 0) ( π 2 , 0 ) , 0) and ( π 2 , π 2 )
Line integrals Use Green’s Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. 32. ∮ C 〈 sin y , x 〉 ⋅ d r , where C is the boundary of the triangle with vertices (0, 0) ( π 2 , 0 ) , 0) and ( π 2 , π 2 )
Solution Summary: The author evaluates the value of the line integral with the help of Green’s Theorem and sketch a graph.
Line integrals Use Green’s Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful.
32.
∮
C
〈
sin
y
,
x
〉
⋅
d
r
, where C is the boundary of the triangle with vertices (0, 0)
(
π
2
,
0
)
, 0) and
(
π
2
,
π
2
)
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.
Evaluate the line integral ∫CF→⋅dr→ using the Fundamental Theorem of Line Integrals if F→(x,y)=(4x+4y)i→+(4x+4y)j→and Cis the smooth curve from (−1,1)to (5,6).
Enter the exact answer.
∫CF→⋅dr→=
No integrals Let F = ⟨2z, z, 2y + x⟩, and let S be the hemisphereof radius a with its base in the xy-plane and center at the origin.a. Evaluate ∫∫S (∇ x F) ⋅ n dS by computing ∇ x F and appealing to symmetry.b. Evaluate the line integral using Stokes’ Theorem to check part (a).
Using Cauchy's Theorem calculate the following integral and the singular points of the function, where
C: z(t) = 3*cost(t) + i*(3+ 3*sin(t)) 0 < t < 2π
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