Notes and course summaries on double integrals, all the rules you need to solve your exercises. The words are in basic English but the math part is easy to grasp.
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≡
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