Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. ∫ C 3 ( x − y ) d s C : r ( t ) = t i + ( 2 − t ) j 0 ≤ t ≤ 1
Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. ∫ C 3 ( x − y ) d s C : r ( t ) = t i + ( 2 − t ) j 0 ≤ t ≤ 1
Solution Summary: The author explains how to calculate the line integral displaystyle undersetCint3(x-y)ds along the path.
Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path.
∫
C
3
(
x
−
y
)
d
s
C
:
r
(
t
)
=
t
i
+
(
2
−
t
)
j
0
≤
t
≤
1
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.
Nonuniform straight-line motion Consider the motion of an object given by the position function r(t) = ƒ(t)⟨a, b, c⟩ + ⟨x0, y0, z0⟩, for t ≥ 0,where a, b, c, x0, y0, and z0 are constants, and ƒ is a differentiable scalar function, for t ≥ 0.a. Explain why r describes motion along a line.b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?
Using the Hough transform i) Develop a general procedure for obtaining the normal representation of a line from its slope-intercept form, y = ax + b. ii) Find the normal representation of the line y = – 2x + 1.
Application of Green's theorem
Assume that u and u are continuously differentiable functions. Using Green's theorem,
prove that
JS
D
Ur
Vy
dA=
u dv,
where D is some domain enclosed by a simple closed curve C with positive orientation.
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