1Substitutions in Multiple Integrals

University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
27
Substitutions in Multiple Integrals
This section shows how to evaluate multiple integrals by substitution. As in single integration, the goal of substitution is to replace complicated integrals by ones that are easier to evaluate. Substitutions accomplish this by simplifying the integrand, the limits of integration, or both.
Substitutions in Double Integrals
Suppose that a region G in the uv -plane is transformed one-to-one into the region R in the xy-plane by equations of the form x = g(u, v) , y = h(u, v)
x = g(u, v)
y = h(u, v)
: The Jacobian determinant , with x = g(u, y) , y = h(u, y)
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
28
In polar coordinates :
we have r and ? in place of u and v .
x = r cos? and y = r sin? , the Jacobian is
, by using u = , v = y/2
EXAMPLE 17: Evaluate
Solution:
v = y/2 y = 2v ,
u = u = x - u = x - x = u+v
? find the boundaries of G by substituting these expressions into the equations for the boundaries of R
? x = = u+v v = u+v u =0
? x = +1 +1 = u+v v +1 = u+v u =1
? y = 0 0 = 2v v = 0
? y = 4 4 = 2v v = 2
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
29
Substitutions in Triple Integrals:
Suppose that a region G in uvw -space is transformed one-to-one into the region D in xyz-space by differentiable equations of the form:
x = g(u, v,w) , y = h(u, v, w) , z = k(u, v, w)
F[g(u, v,w), h(u, v, w), k(u, v, w)] = H(u, v, w)
=
J(u, v, w) : The Jacobian determinant, with x = g(u, y,w) , y = h(u, y,w), z = k(u, v, w)
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
30
In cylindrical coordinates :
The transformation from Cartesian r?z - space to Cartesian xyz-space is given by the equations
x = r cos? and y = r sin? , z =z , the Jacobian is
In spherical coordinates :
The transformation from Cartesian ??? - space to Cartesian xyz-space is given by the equations
x = sin? cos? , y = ? sin? sin? and z= cos? , the Jacobian is
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
31
EXAMPLE 17: Evaluate
by applying the transformation u = v = y/2 w = z/3
Solution:
u = u = x - u = x - x = u+v
v = y/2 y = 2v , z = 3w
? find the boundaries of G by substituting these expressions into the equations for the boundaries of D
? x = = u+v v = u+v u =0
? x = +1 +1 = u+v v +1 = u+v u =1
? y = 0 0 = 2v v = 0
? y = 4 4 = 2v v = 2
? z = 0 0 = 3w w = 0
? z = 3 3 = 3w w = 1
University of Babylon Lecture: Hussein ALbermany
Collage of Material Engineering Subject: Mathematics
Department of Metallurgy Engineering Stage : 2nd stage
32