/sci/ - Science & Math

[Anonymous No.5812168]

Is it true that <span class="math">\{(x, y) \in \mathbb{R}^2 : x \in [0, 2] \text{ and } 1 \le y \le e^x \} = \{(x, y) \in \mathbb{R}^2 : y \in [1, e^2] \text{ and } \ln y \le x \le 2 \}[/spoiler], /sci/?

[Anonymous No.5812171]

>>5812168
Is jsMath failing to load for anyone else?

Anyway, pic related is the reason for my original post. I know that the two integrals are equal, though I just want to know if I'm setting up my working correctly.

[Anonymous No.5812176]

>>5812171
>jsmath
yeah, it's a problem
>>>q/623173

[Anonymous No.5812178]

>>5812176
>>>/q/623173
>datdrunkensyntaxfail

[Anonymous No.5812180]

>>5812168
For the first integral, I get
<div class="math">e^{2} + 3 </div>

For the second integral, I get
<div class="math">e^{2} - 3 </div>

Which certainly aren't equal.

[Anonymous No.5812185]

>>5812180
Are you sure? Wolfram alpha says otherwise.

http://www.wolframalpha.com/input/?i=integrate+%28integrate+1+from+y+%3D+1+to+y+%3D+e%5Ex%29+from+x+%3D+0+to+x+%3D+2

http://www.wolframalpha.com/input/?i=integrate+%28integrate+1+from+x+%3D+ln+y+to+x+%3D+2%29+from+y+%3D+1+to+y+%3D+e%5E2

[Anonymous No.5812188]

>>5812180
Oh wait. Turns out I made a typo. Thanks for spotting it for me.

Do you know if the sets I mention really are equal?

[Anonymous No.5812192]

>>5812185
The picture had the outer integral going from 2 to 0, so it was equal to 3 - e^2

[Test No.5812196]

<span class="math">\frac{shit}{nigger}[/spoiler]

[Anonymous No.5812198]

>>5812188
I'd say yes, because the interval of x, has the value of the interval of y when you take the natural logarithm.

x \in [0,2]
y \in [1,e^{2}]

y \in [ln(1),ln(e^{2})] = y \in [0,2]

Same with the regions.

[Anonymous No.5812206]

>>5812198
Thanks, anon.