• Support PF! Buy your school textbooks, materials and every day products Here!

Double Integral evaluation help

  • Thread starter wilcofan3
  • Start date
  • #1
27
0

Homework Statement



Evaluate the double integral [tex]\int \int_{R} ln(xy) dA[/tex] where [tex]R[/tex] is the rectangle bounded by [tex]x=e, x=e^2,y=1,y=e[/tex].

Homework Equations


[tex]ln (xy) = ln x + ln y[/tex]

The Attempt at a Solution



I was just wondering, do I need to do anything other than take the integral with respect to x, plug in the bounds, and then take the integral with respect to y, and plug in the bounds? I know to use integration by parts for ln x, and I know to break the ln xy up, but is it fine just to treat the ln y as a constant even if I do integration by parts? That would make it become x ln y. Thanks.
 

Answers and Replies

  • #2
33,171
4,858

Homework Statement



Evaluate the double integral [tex]\int \int_{R} ln(xy) dA[/tex] where [tex]R[/tex] is the rectangle bounded by [tex]x=e, x=e^2,y=1,y=e[/tex].

Homework Equations


[tex]ln (xy) = ln x + ln y[/tex]

The Attempt at a Solution



I was just wondering, do I need to do anything other than take the integral with respect to x, plug in the bounds, and then take the integral with respect to y, and plug in the bounds? I know to use integration by parts for ln x, and I know to break the ln xy up, but is it fine just to treat the ln y as a constant even if I do integration by parts? That would make it become x ln y. Thanks.
If you're integrating with respect to x, then yes, ln y can be treated as a constant.

But what is "it" that would become x ln y?
 
  • #3
27
0
If you're integrating with respect to x, then yes, ln y can be treated as a constant.

But what is "it" that would become x ln y?
I was referring to the constant when I integrate with respect to x. Sorry, I was a little vague.

Thank you, though, I just was a little unsure on whether having a "u" substitution would change things up, but I should just treat it as a constant.

I ended up with the result [tex]2e^3 - 2e^2[/tex].
 
  • #4
43
0

Homework Statement



Evaluate the double integral [tex]\int \int_{R} ln(xy) dA[/tex] where [tex]R[/tex] is the rectangle bounded by [tex]x=e, x=e^2,y=1,y=e[/tex].

Homework Equations


[tex]ln (xy) = ln x + ln y[/tex]

The Attempt at a Solution



I was just wondering, do I need to do anything other than take the integral with respect to x, plug in the bounds, and then take the integral with respect to y, and plug in the bounds? I know to use integration by parts for ln x, and I know to break the ln xy up, but is it fine just to treat the ln y as a constant even if I do integration by parts? That would make it become x ln y. Thanks.
Just do the integral with respect to why then x, so it will be written dy dx,

for dy the bounds will just be from 1 to e,

then for dx it will go from e to e^2
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,794
923
[tex]\int_{x=a}^b \int_{y= c}^d f(x)+ g(y) dy dx= \left(\int_{x= a}^b f(x)dx\right)\left(\int_{y= c}^d dy\right)+ \left(\int_{x=a}^b dx\right)\left(\int_{y= c}^d g(y)dy\right)[/tex]
[tex]= (d-c)\int_{x=a}^b f(x)dx+ (b-a)\int_{y= c}^d g(y)dy[/tex]
 

Related Threads for: Double Integral evaluation help

Replies
1
Views
678
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
16
Views
2K
  • Last Post
Replies
2
Views
848
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
7
Views
689
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
1
Views
1K
Top