DOUBLE INTEGRAL INTRODUCTION
In this section we want to integrate a function of two variables,f(x,y). With
functions of one variable we integrated over an interval (i.e. a one-
dimensional space) and so it makes some sense then that when integrating
a function of two variables we will integrate over a region of R2 (two-
dimensional space). It is not possible to take a single integral with respect to
two variables. Instead, to integrate over two variables, two separate integrations
must be performed - one over each variable.
We can easily find the area of a rectangular region by double integration.
The double integral of a function of two variables, say f(x, y) over a
rectangular region can be denoted as: ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝐴 = ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
Double Integral in Polar Coordinates
1. ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝐴 = ∬𝑅 𝑓(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑖𝑛𝜃)𝑟𝑑𝑟𝑑𝜃
𝜃2 𝑟2
≫ ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝐴 = ∫𝜃1 ∫𝑟1 𝑓(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑖𝑛𝜃)𝑟𝑑𝑟𝑑𝜃
Where,
f(r cosθ, r sinθ) = f(r, θ)
Some of the applications of double integration are , it helps in computing
volumes and another application regarding geometry is finding areas of
surfaces.
Evaluate the below integrals.
,
In this section we want to integrate a function of two variables,f(x,y). With
functions of one variable we integrated over an interval (i.e. a one-
dimensional space) and so it makes some sense then that when integrating
a function of two variables we will integrate over a region of R2 (two-
dimensional space). It is not possible to take a single integral with respect to
two variables. Instead, to integrate over two variables, two separate integrations
must be performed - one over each variable.
We can easily find the area of a rectangular region by double integration.
The double integral of a function of two variables, say f(x, y) over a
rectangular region can be denoted as: ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝐴 = ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
Double Integral in Polar Coordinates
1. ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝐴 = ∬𝑅 𝑓(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑖𝑛𝜃)𝑟𝑑𝑟𝑑𝜃
𝜃2 𝑟2
≫ ∬𝑅 𝑓(𝑥, 𝑦)𝑑𝐴 = ∫𝜃1 ∫𝑟1 𝑓(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑖𝑛𝜃)𝑟𝑑𝑟𝑑𝜃
Where,
f(r cosθ, r sinθ) = f(r, θ)
Some of the applications of double integration are , it helps in computing
volumes and another application regarding geometry is finding areas of
surfaces.
Evaluate the below integrals.
,
