Michael Corral: Vector Calculus

3.5 Change of Variables in Multiple Integrals 121
The change of variables formula can be used to evaluate double integrals in polar
coordinates. Letting
x= x(r,θ)=rcosθ and y=y(r,θ)=rsinθ,
we have
∂x ∂x
J(u,v)=
∂r ∂θ
∂y ∂y
=
cosθ −rsinθ
∣
∣ sinθ rcosθ
∂r ∂θ
∣
so we have the following formula:
∣ = rcos2 θ+rsin 2 θ=r⇒|J(u,v)|=|r|=r,
Double Integral in Polar Coordinates
f(x,y)dxdy=
R
R ′ f(rcosθ,rsinθ)rdrdθ, (3.24)
where the mapping x=rcosθ, y=rsinθ maps the region R ′ in the rθ-plane onto the
region R in the xy-plane in a one-to-one manner.
Example 3.10. Find the volume V inside the paraboloid z= x 2 +y 2 for 0≤z≤1.
Solution: Using vertical slices, we see that
V= (1−z)dA= (1−(x 2 +y 2 ))dA,
R
where R={(x,y) : x 2 + y 2 ≤ 1} is the unit disk in 2
(seeFigure3.5.2). Inpolarcoordinates(r,θ)weknow
that x 2 + y 2 = r 2 and that the unit disk R is the set
R ′ ={(r,θ) : 0≤r≤1,0≤θ≤2π}. Thus,
V=
=
=
=
∫ 2π ∫ 1
0 0
∫ 2π ∫ 1
0
∫ 2π
0
∫ 2π
= π 2
0
0
R
(1−r 2 )rdrdθ
(r−r 3 )drdθ
(
r 2 2 − r4 4
1
4 dθ
∣
∣ r=1
r=0
)
dθ
1
z
x 2 +y 2 = 1
y
0
x
Figure 3.5.2 z= x 2 +y 2