Review for Chapter 16
1. Evaluation of double integrals: (x, y coordinates)
Z ZR
f(x, y)dA =Zb
aZg2(x)
g1(x)
f(x, y)dydx, if Ris a≤x≤b, g1(x)≤y≤g2(x).
Z ZR
f(x, y)dA =Zd
cZh2(y)
h1(y)
f(x, y)dydx, if Ris c≤y≤d, h1(y)≤x≤h2(y).
When you can evaluate the integral by either way, you may want to choose a
simpler way.
2. Evaluation of double integrals: (Polar coordinates)
Z ZR
f(x, y)dA =Zb
aZg2(r)
g1(r)
f(rcos θ, r sin θ)rdθdr, if Ris a≤r≤b, g1(r)≤θ≤g2(r).
Z ZR
f(x, y)dA =Zb
aZh2(θ)
h1(θ)
f(rcos θ, r sin θ)rdrdθ, if Ris a≤θ≤b, h1(θ)≤r≤h2(θ).
A useful fact: dA =rdrdθ =dxdy.
2. Some applications of double integrals.
2a. Area of region Ris R RRdA.
2b. The volume beween z=f(x, y) and z=g(x, y) when (x, y) are in Ris
R RR(f(x, y)−g(x, y))dA.
2c. (Applications in Physics) Let ρ(x, y) be the density of lamina whose region is
R. Then the mass and the centroid of the lamina (x, y) is
mass m=R RRρ(x, y)dA
x=1
mR RRxρ(x, y)dA
y=1
mR RRyρ(x, y)dA
3. Evaluation of triple integrals: (rectangular coordinates) The main idea is the
same as the cross section idea. The following gives a way to reduce a triple integral
to a double integral.
ZZZT
f(x, y, z)dV =Z ZR(Zh1(x,y )
h2(x,y)
f(x, y, z)dz )dA if Tis h1(x, y)≤z≤h2(x, y ),(x, y) in R.
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