Change of Variables


by shards5
Tags: variables
shards5
shards5 is offline
#1
Mar25-11, 05:25 PM
P: 38
1. The problem statement, all variables and given/known data
Suppose D is the parallelogram in the xy-plane with vertices P(-1,5), Q(1,-5), R(5,-1), S(3,9)
[tex] \int\int ^{}_{D} (6x+12y) dA [/tex]
HINT: Use transformation x = [tex]\frac{1}{6}[/tex](u+v) and y = [tex]\frac{1}{6}[/tex] (-5u+v).

2. Relevant equations



3. The attempt at a solution
Calculating the Jacobian I get
dx/du = 1/6 dx/dv = 1/6
dy/du = -5/6 dy/dv = 1/6
which gives me
1/36 - (-5/36) = 1/6
From there I need the new intervals of integration.
P-Q
[tex]\frac{5-(-5)}{-1-(1)}[/tex] = 10/-2 = -5
y = 5x + 5 -> plugging in . . .
-5/6u+1/6v = -5/6u - 5/6v + 5 -> u cancels out and then we get 6/6v = 5 which means v = 5
S-R
[tex]\frac{9-(-1)}{3-5}[/tex] = -5
y = -5x + 9
-5/6u+1/6v = -5/6u+5/6v + 9 -> v = 9
R-Q
[tex]\frac{(-1)-(-5)}{5-1}[/tex] = 1
y = x-1 -> -5/6u + 1/6v = 1/6u +1/6v -1 -> u = 1
S-P
[tex]\frac{(9-(5)}{3-(-1)}[/tex] = 1
y = x-9 -> -5/6u +1/6v = 1/6u + 1/6v - 9 -> u = 9

Plugging in the x and y values and the Jacobian into the integral we get the following new integral.
[tex] \int^{9}_{5}\int^{9}_{1} (u+v + (-10u+2v)) * 1/6 dudv [/tex]
which simplifies to . . .
[tex] \int^{9}_{5}\int^{9}_{1} (-4/3u+1/2v) dudv [/tex]
After the first integration I get . . .
[tex] \int^{9}_{5} (-2/3u^2+1/2v*u) dudv [/tex]
Plugging in 9 and 1 I get
-54-9/2v+50/3-5/2v = -37.33333 - 7v
Integrating what I got I get
-37.33333v - 7/2v2
And after plugging in 9 and 5 I get -345.333
which is wrong. So what am I doing wrong?
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes

Register to reply

Related Discussions
Change of Variables - Help! Calculus & Beyond Homework 4
Change of variables Calculus & Beyond Homework 1
change of variables Calculus & Beyond Homework 4
Change of Variables Calculus & Beyond Homework 1
change of variables Introductory Physics Homework 9