Change of Variables: Integrals w/Polar Coordinates

[ver_mathstats]

Homework Statement

Please explain why the integrals yield two different values.

Relevant Equations

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We have two different integrals, the first one being ∫∫e^rdrdθ where -1≤r≤1 and 0≤θ≤π which equals approximately 7 and ∫∫e^rdrdθ where 0≤r≤1 and 0≤θ≤2π which equals approximately 11. Why do these integrals have different values and do not go against the change of variables theorem?

I'm having trouble understanding why these both do not equal each other and how to approach demonstrating this using change of variables. I went over the change of variables for double integrals and polar coordinates but am having trouble connecting the two. Any hint would be appreciated, thank you.

 

[Orodruin]

Because $e^r$ is not an even function.

 

[FactChecker]

I don't see that this is a problem involving a change of variables at all. Since $e^r$ is not a function of $\theta$, $\int_{0}^{\pi}\int_{-1}^{1}e^r drd\theta = \pi \int_{-1}^{1}e^r dr$
and
$\int_{0}^{2\pi}\int_{0}^{1}e^r drd\theta = 2\pi \int_{0}^{1}e^r dr$
So you are essentially asking why
$\int_{-1}^{1}e^r dr \ne 2 \int_{0}^{1}e^r dr$