Show that the integral is independent of the path, and use Theorem 15.3.1 to find its value.
∫
−
1
,
2
0
,
1
3
x
−
y
+
1
d
x
−
x
+
4
y
+
2
d
y
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.
Calculate (4x² + 5e³) dy, where C is the triangle of vertices (0,0), (2,0) and (2,2).
[(4x²+
Note: For a triangle, parameterize each side as a line segment, and calculate the line integral on
that side. Then, add up all of the answers for each side to find the final answer. Make sure to go in
order, from the first point to the second point, then from the second point to the third point, and then
from the third point back to the first point, because the order does matter.
5e) dy =
Assume that a rocket is in space and is following the path y = – 6(x − 7)² + 3, from left to right where
the origin is set to be a certain point in space.
At what point on that path should the engine be turned off in order to coast along the tangent line to
another ship that is not moving relative to the origin of the path and is at the point (10, 3)?
(Once the engine is turned off, there is no force pushing it in a certain direction, so it continues along the
same path that the velocity is pointing at that specific time. This means it continues along a path that is a
tangent line to the original curve.)
Note: The graph really helps on this one. Go ahead and draw it and you will see that it helps. It's all about
the picture on this one!
Coordinates:
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