Lecture on Multiple Integrals

DOUBLE INTEGRALS AND GREEN’S THEOREM IN A PLANE

, Area of a Volume of a
DOUBLE INTEGRALS Region Solid
AND GREEN’S
THEOREM IN A PLANE Presentation
outlines
LEARNING UNIT 3 Centroid and
Green’s
Moment of
Theorem
Inertia

, DOUBLE INTEGRATION
• In a double integral we integrate a
function 𝑓(𝑥, 𝑦) , called the
integrand, over a closed bounded
region ℛ 2 in the 𝑥𝑦 −plane, whose
boundary curve has a unique
tangent at each point.
• double integral of f(x, y) over the
region ℛ, and is denoted by


ඵ 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 𝑜𝑟 ඵ 𝑓 𝑥, 𝑦 𝑑𝐴
ℛ ℛ

, PROPERTIES OF DOUBLE INTEGRALS

ඵ 𝑘 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = 𝑘 ඵ 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 , 𝑘 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
ℛ ℛ



ඵ 𝑓 𝑥, 𝑦 + 𝑔(𝑥, 𝑦) 𝑑𝑥𝑑𝑦 = ඵ 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 + ඵ 𝑔 𝑥, 𝑦 𝑑𝑥𝑑𝑦
ℛ ℛ ℛ



ඵ 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = ඵ 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦 + ඵ 𝑓 𝑥, 𝑦 𝑑𝑥𝑑𝑦
ℛ1 +ℛ2 ℛ1 ℛ2