Integrating Double Integral of Cos(x+y) from 0 to pi: Step-by-Step Guide

[Math10]

Homework Statement

How to integrate the double integral cos(x+y) dy dx from 0 to pi and from 0 to pi again.

Homework Equations

The answer is -4.

The Attempt at a Solution

Here's the work:
u=x+y
du=dy
cos(u)du=sin(x+y)
integral of [sin(x+y)] evaluate from 0 to pi dx from 0 to pi
integral of (sin(x+pi)-sin(x))dx from 0 to pi
And what's next?

 

[LCKurtz]

Homework Statement

How to integrate the double integral cos(x+y) dy dx from 0 to pi and from 0 to pi again.

Homework Equations

The answer is -4.

The Attempt at a Solution

Here's the work:
u=x+y
du=dy
cos(u)du=sin(x+y)
integral of [sin(x+y)] evaluate from 0 to pi dx from 0 to pi
integral of (sin(x+pi)-sin(x))dx from 0 to pi
And what's next?

Integrate again, similar to what you did for the first integral.

 

[Fightfish]

Er...doesn't the final step that you list give you the answer already?

 

[Math10]

But I can't. Because sin(x+pi)-sin(x)=sin(x)+sin(pi)-sin(x)=sin(pi)=0. The integral of 0 is?

 

[LCKurtz]

But I can't. Because sin(x+pi)-sin(x)=sin(x)+sin(pi)-sin(x)=sin(pi)=0. The integral of 0 is?

$\sin(a+b)\ne \sin a +\sin b$

 

[Fightfish]

Because sin(x+pi)=sin(x)+sin(pi)

That is most certainly not correct. Can you check your compound angle formula?

 

[Math10]

Then what's sin(a+b)?

 

[Fightfish]

http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html

 

[Math10]

@Fightfish , thank you so much for the info! I got it!