Double Integral 1
• Iterated Integral and Fubini’s Theorem
for a region of type I,
R = {(x, y) | a ≤
x ≤ b, p(x)
≤ y ≤ q(x)}.
Figure
for a region of type II,
R = {(x, y) | u(y) ≤ x ≤ v(y),
c ≤ y ≤ d}.
Figure
• Double Integral over
Rectangular Regions
If R is the rectangular
region [a, b] × [c, d], then
In the special case where the
integrand f(x, y) can be written as g(x, y) h(y)
we have
• Change of Variables
Is the jacobian formations (x, y) →
(u, v), and S is the pullback of R which can be computed
by x = x (u, v), y = y (u, v)
into the definition of R.
• Polar Coordinater
x = r cos θ, y = r sin θ
Figure
• Double Integral in Polar
Coordinates
The Differential dxdy for Polar Coordinates is
Let the region R is determined as follows:
0 ≤ g(θ) ≤ h(θ), α ≤ θ ≤ β, where β – α ≤ 2π.
Then
Figure
If the region R is the polar rectangle given
by
0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β,
where β – α ≤ 2π.
Figure
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