For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals. 114. [ T] ∮ c [ arctan y x − x y x 2 + y 2 ] d x + [ x 2 x 2 + y 2 + e − y ( 1 − y ) ] d y , where C is any smooth curve from (1, 1) to (-1,2)
For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals. 114. [ T] ∮ c [ arctan y x − x y x 2 + y 2 ] d x + [ x 2 x 2 + y 2 + e − y ( 1 − y ) ] d y , where C is any smooth curve from (1, 1) to (-1,2)
For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.
114.
[
T]
∮
c
[
arctan
y
x
−
x
y
x
2
+
y
2
]
d
x
+
[
x
2
x
2
+
y
2
+
e
−
y
(
1
−
y
)
]
d
y
,
where C is any smooth curve from (1, 1) to (-1,2)
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.
3. Consider the function: f (x) = v16 – x² + 2
a. Sketch the graph.
b. Using geometric formulas (areas), compute the integral: (V16 – x² + 2) dx
Use Green's Theorem to evaluate the line integral of F = (x6, 3x)
around the boundary of the parallelogram in the following figure (note the orientation).
(xo.)
(X0.0)
Sex6 dx + 3x dy
=
(2x-Y)
·x
With xo =
7 and yo
=
7.
|-4 Evaluate the line integral by two methods: (a) directly and
(b) using Green's Theorem.
I. $. (x – y) dx + (x + y) dy,
C is the circle with center the origin and radius 2
-
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