Evaluating a Line Integral in Differential Form In Exercises 57–64, evaluate ∫ C ( 2 x − y ) d x + ( x + 3 y ) d y . C: line segments from (0, 0) to (0,-3) and (0,-3) to (2,-3).
Evaluating a Line Integral in Differential Form In Exercises 57–64, evaluate ∫ C ( 2 x − y ) d x + ( x + 3 y ) d y . C: line segments from (0, 0) to (0,-3) and (0,-3) to (2,-3).
Solution Summary: The author explains that the line segment C can be composed of two parts C 1 and C 2.
In Exercises 57–64, evaluate
∫
C
(
2
x
−
y
)
d
x
+
(
x
+
3
y
)
d
y
.
C: line segments from (0, 0) to (0,-3) and (0,-3) to (2,-3).
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
A driver is traveling along a straight road when a buffalo runs into the street. This driver has a reaction time of 0.75 seconds. When the driver sees the buffalo he is traveling at 44 ft/s, his car can decelerate at 2 ft/s^2 when the brakes are applied. What is the stopping distance between when the driver first saw the buffalo, to when the car stops.
Topic 2
Evaluate S
x
dx, using u-substitution. Then find the integral using
1-x2
trigonometric substitution. Discuss the results!
Topic 3
Explain what an elementary anti-derivative is. Then consider the following
ex
integrals: fed dx
x
1
Sdx
In x
Joseph Liouville proved that the first integral does not have an elementary anti-
derivative Use this fact to prove that the second integral does not have an
elementary anti-derivative. (hint: use an appropriate u-substitution!)
1. Given the vector field F(x, y, z) = -xi, verify the relation
1
V.F(0,0,0) = lim
0+ volume inside Se
ff F• Nds
SE
where SE is the surface enclosing a cube centred at the origin and having edges of length 2€. Then,
determine if the origin is sink or source.
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