31–40. Line integrals Use Green’s Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. ∮ C 1 1 + y 2 d x + y d y , where C is the boundary of the triangle with vertices (0, 0), (1, 0), and (1, 1)
31–40. Line integrals Use Green’s Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. ∮ C 1 1 + y 2 d x + y d y , where C is the boundary of the triangle with vertices (0, 0), (1, 0), and (1, 1)
31–40. Line integrals Use Green’s Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful.
∮
C
1
1
+
y
2
d
x
+
y
d
y
, where C is the boundary of the triangle with vertices (0, 0), (1, 0), and (1, 1)
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.
Use Green's theorem to evaluate
F. dr. (Check the orientation of the curve before applying the theorem.)
F(x, y) = (y cos(x) – xy sin(x), xy + x cos(x)), Cis the triangle from (0, 0) to (0, 10) to (2, 0) to (0, 0)
Find the points on the graph of z = xy' + 8y¯' where the tangent plane is parallel to 9x + 5y + 9z = 0.
%3D
%3D
(Give your answer as a comma-separated list of points in the form (*, *, *). Express numbers in exact form. Use symbolic
notation and fractions where needed.)
point(s):
WHite the veD secsand orde equation as is equivalent svstem of hirst order equations.
u" +7.5z - 3.5u = -4 sin(3t),
u(1) = -8,
u'(1)
-6.5
Use v to represent the "velocity fumerion", ie.v =().
Use o and u for the rwo functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.)
+7.5v+3.5u-4 sin 3t
Now write the system using matrices:
dt
3.5
7.5
4 sin(3t)
and the initial value for the vector valued function is:
u(1)
v(1)
3.5
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