Multiple Integrals

In this chapter we consider the integral of a function of two variables ƒ(x, y) over a region in the plane and the integral of a function of three variables ƒ(x, y, z) over a region in space. We can use multiple integrals to calculate quantities that vary over two or three dimensions, such as the total mass or the angular momentum of an object of varying density and the volumes of solids with general curved boundaries.We begin our investigation of double integrals by considering the simplest type of planar region, a rectangle. We consider a function ƒ(x, y) defined on a rectangular region R , given by
R: a ? x ? b , c ? y ? d.
?Ak = ?xk ?yk
Double Integrals as Volumes
When ƒ(x, y) is a positive function over a rectangular region R in the xy-plane, we may interpret the double integral of ƒ over R as the volume of the 3-dimensional solid region over the xy-plane bounded below by R and above by the surface z = ƒ(x, y)
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
2
Properties of Double Integrals
THEOREM 1 Fubini’s Theorem (First Form)
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
3
EXAMPLE 1: Calculate the volume under the plane over the rectangular region R: 0 ? x ? 2 , 0 ? y ? 1 ?
Solution:
EXAMPLE2:
Solution