Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. ∫ C ( x 2 + y 2 + z 2 ) d s C : r ( t ) = sin t i + cos t j + 2 k 0 ≤ t ≤ π 2
Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path. ∫ C ( x 2 + y 2 + z 2 ) d s C : r ( t ) = sin t i + cos t j + 2 k 0 ≤ t ≤ π 2
Solution Summary: The author explains how to calculate the line integral of displaystyle
Evaluating a Line Integral In Exercises 19-22, evaluate the line integral along the given path.
∫
C
(
x
2
+
y
2
+
z
2
)
d
s
C
:
r
(
t
)
=
sin
t
i
+
cos
t
j
+
2
k
0
≤
t
≤
π
2
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.
Using Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)
A. State the Fundamental Theorem of Calculus for Line Integrals.
B. Let f(x, y, z) = x^2 + 2y^2 + 3z^2 and F = grad f. Find the line integral of F along the line C with parametric equations
x = 1 + t, y = 1 + 2t, z = 1 + 3t, 0 ≤ t ≤ 1.
You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.
Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 4)
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