# Thread: Integrals

1. ## Integrals

Hello! I should determine the values of the following integrals without really calculating:

(1) $\int \limits_{B(2, \pi)}^{} \exp(x) \cdot \cos(y) dx dy$

(2) $\int \limits_{\partial B(x_0, \epsilon)}^{} \log(|x|) dS(x)$, with $0<\epsilon<|x_0|, x_0 \in \mathbb{R}^2$.

I do not have any clue how to solve this exercise. How can one see the values of these integrals?

Can somebody please help me?

2. Does really nobody have an idea? Or a hint? I am completely stuck . I would be thankful for any answers...

Greetings

3. Originally Posted by gammafunction
Hello! I should determine the values of the following integrals without really calculating:

(1) $\int \limits_{B(2, \pi)}^{} \exp(x) \cdot \cos(y) dx dy$

(2) $\int \limits_{\partial B(x_0, \epsilon)}^{} \log(|x|) dS(x)$, with $0<\epsilon<|x_0|, x_0 \in \mathbb{R}^2$.

I do not have any clue how to solve this exercise. How can one see the values of these integrals?

Can somebody please help me?
You could help us by stating what you want $B(2, \pi)$ to denote.

CB

4. Originally Posted by gammafunction
Does really nobody have an idea? Or a hint? I am completely stuck . I would be thankful for any answers...

Greetings
If you have recieved no response you should consider providing more information or clarification for your question.

Bumping, which is what this post is, is against the rules here and in this case unnecessary you have not recieved an answer because your notation for the volume (and surface) over which the integrals are to be evaluated are not clear. So just provide more explanation in a follow-up post.

CB

5. Sorry, i did not want to be impatient, i was just interested . And sorry for that mistake, it should of course be the ball around $(2,\pi)$ with radius r, i will change that.

But i have a solution: as both functions are harmonic (log|x| being the real part of a holomorphic function) one can use the mean-value theorem which gives the following values for the integrals:
$\int \limits_{B((2, \pi),r)}^{} \exp(x) \cdot \cos(y) dx dy=-r^2 \pi \exp(2)$ and

$\int \limits_{\partial B(x_0, \epsilon)}^{} \log(|x|) dS(x)=2r \pi \log(|x_0|)$

Greetings