Therefore ,E = xex2 + y2 + z2 dv, where e is the portion of the unit ball x2 + y2 + z2 ≤ 1 that lies in the first octant whose value is π/8
The volume of an inhabited three-dimensional space is calculated. It is commonly measured in a variety of imperial, US-standard, and SI-derived units. Volume and the notion of length are interconnected.
Here,
Given :
portion of unit ball = [tex]xe^{x^{2}+y^{2} +z^{2} }[/tex]
and x2 + y2 + z2 ≤ 1
So, using this
We find the triple integral or the volume integral of the expression = [tex]xe^{x^{2}+y^{2} +z^{2} }[/tex]
=> [tex]\int\int\int_{e} xe^{x^{2}+y^{2} +z^{2} }[/tex]dV
=> [tex]\int\limits^{\alpha = \pi /2}_{\alpha =0}\int\limits^{\beta = \pi /2}_{\beta =0} \int\limits^{p= 1}_{p=0}[/tex] pcos[tex]\beta[/tex]sin[tex]\alpha[/tex][tex]e^{{p}^{2}[/tex] [tex]p^{2}[/tex]sin[tex]\alpha[/tex] dpd[tex]\alpha[/tex]d[tex]\beta[/tex]
=>[tex]\int\limits^{\alpha = \pi /2}_{\alpha =0}[/tex][tex]sin^{2}\alpha[/tex] d[tex]\alpha[/tex] + [tex]\int\limits^{\beta = \pi /2}_{\beta =0}[/tex]cos[tex]\beta[/tex] d[tex]\beta[/tex] + [tex]\int\limits^{p= 1}_{p=0} p^{2} e^{p^{2}}[/tex] dp
=>π/8
Therefore ,E xex2 + y2 + z2 dv, where e is the portion of the unit ball x2 + y2 + z2 ≤ 1 that lies in the first octant whose value is π/8
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