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Question-215704




Question Number 215704 by universe last updated on 15/Jan/25
Commented by universe last updated on 16/Jan/25
HI MR W please solve this problem
HIMRWpleasesolvethisproblem
Answered by mr W last updated on 16/Jan/25
∫∫∫(x^2 +y^2 +z^2 )dxdydz  =∫_0 ^a ∫_0 ^(a−z) ∫_0 ^(a−z−y) (x^2 +y^2 +z^2 )dxdydz  =∫_0 ^a ∫_0 ^(a−z) [(((a−z−y)^3 )/3)+y^2 (a−z−y)+z^2 (a−z−y)]dydz  =∫_0 ^a ∫_0 ^(a−z) [(((a−z−y)^3 )/3)+(a−z)y^2 −y^3 +z^2 (a−z)−z^2 y]dydz  =∫_0 ^a [(((a−z)^4 )/(3×4))+(((a−z)(a−z)^3 )/3)−(((a−z)^4 )/4)+z^2 (a−z)(a−z)−((z^2 (a−z)^2 )/2)]dz  =∫_0 ^a [(((a−z)^4 )/6)+((z^2 (a−z)^2 )/2)]dz  =∫_0 ^a [(((a−z)^4 )/6)+((z^4 −2az^3 +a^2 z^2 )/2)]dz  =(1/6)×(a^5 /5)+(1/2)((a^5 /5)−((2a×a^4 )/4)+((a^2 ×a^3 )/3))  =(a^5 /(20)) ✓
(x2+y2+z2)dxdydz=0a0az0azy(x2+y2+z2)dxdydz=0a0az[(azy)33+y2(azy)+z2(azy)]dydz=0a0az[(azy)33+(az)y2y3+z2(az)z2y]dydz=0a[(az)43×4+(az)(az)33(az)44+z2(az)(az)z2(az)22]dz=0a[(az)46+z2(az)22]dz=0a[(az)46+z42az3+a2z22]dz=16×a55+12(a552a×a44+a2×a33)=a520
Commented by universe last updated on 16/Jan/25
thank you very much sir
thankyouverymuchsir
Commented by universe last updated on 16/Jan/25
   x= au , y=av,z = aw     ∫∫∫a^2 (u^2 +v^2 +w^2 )a^3 dudvdw  a^5 ∫∫∫u^2 dudvdw +a^5 ∫∫∫v^2 dudvdw+a^5 ∫∫∫w^2 dudvdw  3a^5 ∫∫∫u^(3−1) v^(1−1) w^(1−1) dudvdw   by derichlet integral  3a^5 (( ⌈3⌈1⌈1)/(⌈(3+1+1+1))) = 3a^5 ((2!)/(5!)) = a^5 /20
x=au,y=av,z=awa2(u2+v2+w2)a3dudvdwa5u2dudvdw+a5v2dudvdw+a5w2dudvdw3a5u31v11w11dudvdwbyderichletintegral3a5311(3+1+1+1)=3a52!5!=a5/20
Commented by universe last updated on 16/Jan/25
MR  W sir what is mistake in this solution??
MRWsirwhatismistakeinthissolution??
Commented by mr W last updated on 16/Jan/25
there is no mistake in this solution.  i made a mistake. now it′s fixed.  i also get (a^5 /(20)).
thereisnomistakeinthissolution.imadeamistake.nowitsfixed.ialsogeta520.
Commented by universe last updated on 16/Jan/25
thank you sir
thankyousir
Commented by ajfour last updated on 17/Jan/25
https://youtu.be/KroW0v7N_Jg?si=8W3jW8TXO6USpeV1

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