How could you express the following double integral in term of a single integral?

Question

How could I express the $$\int\int e^{(x^2+y^2)^2} dA$$ in terms of a single integral with respect to r where D is a disk with center (0,0) and radius 1

polar coordinates: r²=x²+y² (Don't forget the extra r in front of the differential). — imranfat, Oct 30 '13 at 18:11

score 5 · Accepted Answer · answered Oct 30 '13 at 18:12

5

Change to polar coordinates:

$$x=r\cos t\;,\;\;y=r\sin t\;,\;\;0\le t\le 2\pi \implies$$

$$\int\int\limits_D e^{(x^2+y^2)^2}dA=\int\limits_0^1\int\limits_0^{2\pi}re^{r^4} dt\,dr=2\pi\int\limits_0^1re^{r^4}dr$$

answered Oct 30 '13 at 18:12

DonAntonio

211,718
17
136
287

2

Nicely done! +1 – amWhy Oct 31 '13 at 12:09

score 2 · Answer 2 · edited Oct 30 '13 at 18:13

2

Since $x^2+y^2=r^2$ and $dxdy\to rdrd\theta$ so you'll have $$2\pi\int_0^1\exp(r^4)rdr$$

edited Oct 30 '13 at 18:13

Adam Saltz

2,596

answered Oct 30 '13 at 18:10

Mikasa

67,374

Yes, sir. +1... – DonAntonio Oct 31 '13 at 05:28
Short and sweet! +1 – amWhy Oct 31 '13 at 12:08
How are you today? I'm having a slow day at MSE, but otherwise I'm doing pretty well. – amWhy Oct 31 '13 at 17:43
@amWhy: Like me. I have been sooooo slow these days also. I am on a local trip nearby. – Mikasa Nov 01 '13 at 17:58

score 1 · Answer 3 · answered Oct 30 '13 at 18:12

1

Hint: Use polar coordinates. There will be no $\theta$ in the new integrand, so we can integrate with respect to $\theta$ immediately.

answered Oct 30 '13 at 18:12

Cameron Buie

102,994

How could you express the following double integral in term of a single integral?

3 Answers3