How can I use the properties of sin(x) and symmetry to understand this integral?

Question

This is a true or false question as review for an upcoming exam. My professor states that this integral holds true using symmetry and the fact that sin(y) is an odd function. Can someone please explain how this holds true using those two concepts? I can't wrap my head around how I could ever use symmetry and sin(y) being an odd function to answer this.

T OR F:

$$ \int_{-1}^{1}\int_0^{1} e^{x^2+y^2}sin(y)dxdy=0$$

See https://math.stackexchange.com/questions/1230999/definite-integral-of-an-odd-function-is-0-symmetric-interval — Vasili, Nov 14 '23 at 20:47
Start by factorizing the double integral as $\int_0^1e^{x^2}\mathrm{d}x\int_{-1}^1e^{y^2}\sin y\mathrm{d}y$. — J.G., Nov 14 '23 at 20:53
@anonymous For the task at hand, that would be counterproductive. — J.G., Nov 14 '23 at 21:36
You’re right, I just saw it .. due to the $\sin(y)$, right? @J.G. — homosapien, Nov 14 '23 at 21:36

score 0 · Answer 1 · answered Nov 14 '23 at 20:49

0

Imagine plotting the odd function and doing the area to the axis. The part beneath has negative area and the part above has positive area.

if it's done between $1$ and $-1$ the areas cancel to make $0$.
Here is a plot of the odd function $y=\sin(x)$ from Desmos, to illustrate

The more complicated equation still has the 'odd' symmetry, negative values have the opposite sign of the positive values

answered Nov 14 '23 at 20:49

John Hunter

611

This makes more sense. So essentially, as long as the 'odd' symmetry exists within the integral and bounds are from a to -a, the integral will always result in 0? – user1245776 Nov 14 '23 at 20:53
Also, what if I had an even function alongside an odd function? Does this still hold true? – user1245776 Nov 14 '23 at 20:54
Yes, the even parts of the equation don't matter as long as it's odd overall, maybe just watch out for two odd functions multiplied, because then it could go even again – John Hunter Nov 14 '23 at 20:55
to check it's odd overall imagine putting two values in e.g. p and -p, if they result in opposite signed values, it'll count as odd – John Hunter Nov 14 '23 at 20:57
I understand. Thank you so much! – user1245776 Nov 14 '23 at 21:04
Ok, all the best – John Hunter Nov 14 '23 at 21:05

How can I use the properties of sin(x) and symmetry to understand this integral?

1 Answers1