Differential Element dA
Differential Element dA
Integration over the region in the plane between the graphs of two continuous functions is performed by setting up type I or type II domains in Cartesian coordinates or as a polar domain in polar coordinates. Double integrals require a student to visualize the differential element of area covering the desired domain of integration. Using an integrand function of value 1 everywhere generates a double integral whose value is the area of the domain.
A plane region is type I if it lies between the graphs of two continuous functions and of on [, ], that is, . Vertical strips (x)(x)f(x,y)dy are integrated as (x)(x)f(x,y)dydx shown by "x strip history" control.
D
g
1
g
2
x
a
b
D={(x,y)|a≤x≤b,(x)≤y≤(x)}
g
1
g
2
g
2
∫
g
1
b
∫
a
g
2
∫
g
1
A plane region is type II if it lies between the graphs of two continuous functions and of on [, ], that is, . Horizontal strips (y)(y)f(x,y)dx are integrated as (y)(y)f(x,y)dxdy shown by "y strip history" control.
D
h
1
h
2
y
c
d
D={(x,y)|c≤y≤d,(y)≤x≤(y)}
h
1
h
2
h
1
∫
h
0
d
∫
c
h
1
∫
h
0
When the domain consists of two type II domains as shown in the plot, a transition occurs from one domain to the other where the integral over the first domain is added to the integral over the second domain. , where , , and are continuous on [, ].
D={(x,y)|(c≤y≤and(y)≤x≤(y))or(≤y≤dand(y)≤x≤(y))}
d
1
h
0
h
1
d
1
h
0
h
2
h
0
h
1
h
2
c
d
For a polar wedge the domain is defined by , where and are continuous on . Radial strips (θ)(θ)f(ρ,θ)ρdρ are integrated as (θ)(θ)f(ρ,θ)ρdρdθ shown by "wedge history" control.
D={(ρ,θ)|α≤θ≤β,(θ)≤ρ≤(θ)}
ρ
1
ρ
2
ρ
1
ρ
2
[α,β]
ρ
2
∫
ρ
1
β
∫
α
ρ
2
∫
ρ
1