Multiple_Integrals_Double_Integrals_Licence_3_Mthématiques

Multiple Integrals

Multiple integrals refer to the integration of functions of multiple variables over multiple regions of space.
In particular, they involve integrating a function of two or more variables over a two-dimensional region, a
three-dimensional region, or a higher-dimensional region.

For example, the double integral of a function 𝑓(𝑥, 𝑦) over a region ℝ in the 𝑥𝑦-plane can be thought of as
finding the volume under the surface 𝑧 = 𝑓(𝑥, 𝑦) and above the region ℝ. Similarly, the triple integral of a
function 𝑓(𝑥, 𝑦, 𝑧) over a region in three-dimensional space can be thought of as finding the volume under
the surface 𝑠 = 𝑓(𝑥, 𝑦, 𝑧) and above the region.

Multiple integrals are a fundamental tool in calculus, and they have many applications in physics,
engineering, economics, and other fields.

Let’s start with some maths now.

I. Double integral :

Let : 𝑓 : 𝐷 ⟼ ℝ2
(𝑥, 𝑦) ⟼ 𝑧 = 𝑓(𝑥, 𝑦)

a continuous and integrable function on 𝐷.

The double integral is denoted by ∬𝑫 𝒇(𝒙, 𝒚). 𝒅𝒙𝒅𝒚 and physically represents the value of the
volume of the cylinder with base D and ℎ𝑒𝑖𝑔ℎ𝑡 = 1 covered by the surface 𝑧 = 𝑓(𝑥, 𝑦).

▪ Let : 𝐷 = 𝐷1 ∪ 𝐷2 with 𝐷1 ∩ 𝐷2 = 𝑠𝑖𝑚𝑝𝑙𝑒 𝑐𝑢𝑟𝑣𝑒 (i.e, a curve whose points are neither in 𝐷1
nor in 𝐷2 ).
𝐷≡


Then : ∬𝑫 𝒇(𝒙, 𝒚). 𝒅𝒙𝒅𝒚 = ∬𝑫 𝒇(𝒙, 𝒚). 𝒅𝒙𝒅𝒚 + ∬𝑫 𝒇(𝒙, 𝒚). 𝒅𝒙𝒅𝒚
𝟏 𝟐


▪ If : 𝑓 ≥ 0 and ∬ 𝑓(𝑥, 𝑦). 𝑑𝑥𝑑𝑦 = 0 then 𝑓(𝑥, 𝑦) = 0 , ∀(𝑥, 𝑦) ∈ 𝐷.

❖ The mean value theorem :

If : 𝑓 is integrable over 𝐷

Then : ∃(𝑐1 , 𝑐2 ) ∈ 𝐷 such that : ∬ 𝒇(𝒙, 𝒚)𝒅𝒙𝒅𝒚 =f(𝒄𝟏 , 𝒄𝟐 ) × 𝒂𝒓𝒆𝒂(𝑫)

❖ Fubini theorem :

Let 𝑓 continuous on 𝐷 = {(𝑥, 𝑦) ∈ ℝ2 ; 𝑎 ≤ 𝑥 ≤ 𝑏 ; 𝜓1 (𝑥) ≤ 𝑦 ≤ 𝜓2 (𝑥)}.

𝒃 𝝍𝟏 (𝜻)
∬ 𝒇(𝒙, 𝒚). 𝒅𝒙𝒅𝒚 = ∫ (∫ 𝒇(𝒙, 𝒚)𝒅𝒚) 𝒅𝒙
𝑫 𝒂 𝝍𝟐 (𝜻)


Same for : 𝐷 = {(𝑥, 𝑦) ∈ ℝ2 ; 𝑐 ≤ 𝑦 ≤ 𝑑 ; 𝜑1 (𝑦) ≤ 𝑥 ≤ 𝜑2 (𝑦)}.