Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem 14.11: ▿ · ( u F ) = ▿ u · F + u ( ▿ · F ). a. Integrate both sides of this identity over a solid region D with a closed boundary S and use the Divergence Theorem to prove an integration by parts rule: ∭ D u ( ∇ ⋅ F ) d V = ∬ S u F ⋅ n d S − ∭ D ∇ u ⋅ F d V . b. Explain the correspondence between this rule and the integration by parts rule for single-variable functions. c. Use integration by parts to evaluate ∭ D ( x 2 y + y 2 z + z 2 x ) d V , where D is the cube in the first octant cut by the planes x = 1, y = 1, and z = 1.
Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem 14.11: ▿ · ( u F ) = ▿ u · F + u ( ▿ · F ). a. Integrate both sides of this identity over a solid region D with a closed boundary S and use the Divergence Theorem to prove an integration by parts rule: ∭ D u ( ∇ ⋅ F ) d V = ∬ S u F ⋅ n d S − ∭ D ∇ u ⋅ F d V . b. Explain the correspondence between this rule and the integration by parts rule for single-variable functions. c. Use integration by parts to evaluate ∭ D ( x 2 y + y 2 z + z 2 x ) d V , where D is the cube in the first octant cut by the planes x = 1, y = 1, and z = 1.
Solution Summary: The author explains the correspondence between the product rule and the integration by parts rule.
Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem 14.11: ▿ · (uF) = ▿u·F + u(▿·F).
a. Integrate both sides of this identity over a solid region D with a closed boundary S and use the Divergence Theorem to prove an integration by parts rule:
∭
D
u
(
∇
⋅
F
)
d
V
=
∬
S
u
F
⋅
n
d
S
−
∭
D
∇
u
⋅
F
d
V
.
b. Explain the correspondence between this rule and the integration by parts rule for single-variable functions.
c. Use integration by parts to evaluate
∭
D
(
x
2
y
+
y
2
z
+
z
2
x
)
d
V
, where D is the cube in the first octant cut by the planes x = 1, y = 1, and z = 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.
Line integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful.
The flux line integral of F = ⟨ex - y, ey - x⟩, where C is theboundary of {(x, y): 0 ≤ y ≤ x, 0 ≤ x ≤ 1}
Use Stokes' Theorem to evaluate (integral) F (dot) dr. C is oriented counterclockwise as viewd from above.
F (x,y,z) = < 2z + x , y - z , x + y >
C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1).
A. State the F undamental Theorem of Calculus for Line Integrals.
B. Let f(x, y, z) = xy + 2yz + 3zx and F = grad f. Find the line integral of F along the line C with parametric equations
x = t, y = t, z = 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.
Thomas' Calculus: Early Transcendentals (14th Edition)
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