Wikipedia:Reference desk/Archives/Mathematics/2012 September 22#Commutability of integration

[http://s1172.photobucket.com/albums/r574/englishwikipedia/?action=view¤t=IMG_0083_zps2be7fe73.jpg problem 20]. I saw the answer key but still unable to understand what is going on. I understand the binary expansion part. I get lost in the statement "it follows that the coefficient of x^2012 is equal to 2^0 + 2^1 + 2^5 = 2^6. [http://s1172.photobucket.com/albums/r574/englishwikipedia/?action=view¤t=IMG_0081_zps19f3eb5a.jpg answer key part 1], [http://s1172.photobucket.com/albums/r574/englishwikipedia/?action=view¤t=IMG_0080_zpsde051df7.jpg answer key part 2]. Thanks!Pendragon5 (talk) 00:02, 22 September 2012 (UTC)

:A product of sums is the sum of all the possible products; say (a+b)*(c+d)*(e+f): the result will contain every product you can make by picking one of each sum,so for example a*d*e, b*c*e, b*c*f and five others.

:In your polynomial, you know which factors supply the power of x part, so the rest must provide the coefficient: showing only the parts that make up the x²⁰¹² term : (x¹⁰²⁴+..)*(x⁵¹²+..)*(x²⁵⁶+..)*(x¹²⁸+..)*(x⁶⁴+..)*(..+ 32)*(x¹⁶+..)*(x⁸+..)*(x⁴+..)*(..+2)*(..+1). And 1*2*32=64=2⁶ Ssscienccce (talk) 02:11, 22 September 2012 (UTC)

::Oh okay I think I got it now. Thanks! I also have another question. How can I convert 2012 into binary number without the answer key?Pendragon5 (talk) 19:37, 22 September 2012 (UTC)

:::Binary_numeral_system#Conversion_to_and_from_other_numeral_systems, first paragraph. — Quondum 21:02, 22 September 2012 (UTC)

::::For a number like 2012, close to 2048, (=2¹¹), a shorter way is: you know that 2¹¹-1 = 2047 is 11111111111, and 2047-2012=35; So you have to subtract 35 which is expressed as a sum of powers of two: 32 + 2 + 1, so from (32;16;8;4;2;1) you know the positions that have to become zeros to get 2012: 11111011100 Ssscienccce (talk) 08:56, 25 September 2012 (UTC)

Is $\backslash int\; (\backslash int\; f(r)\; dr)\; dv$ the same as $\backslash int\; (\backslash int\; f(r)\; dv)\; dr$? 71.207.151.227 (talk) 21:04, 22 September 2012 (UTC)

:See Fubini's theorem. Sławomir Biały (talk) 21:09, 22 September 2012 (UTC)