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Sagot :
The value of the expression on evaluation is 96π / 5.
Here we have to find the value of the expression.
∫∫∫x² dV
E is indeed the solid that exists inside the cylinder x² + y² = 4 above the surface z =0.
E's precession through into xy-plane is x² + y² ≤4 where r≤2 with Ф[0, 2π].
∫∫∫ x² dV = ∫[tex]\int\limits^2\pi _0 {} \, \int\limits^2_0{} \, \int\limits^3r_0[/tex] r² cos² Фrd dФ
= [tex]\int\limits^2\pi _0 {} \, \int\limits^2_0 \int\limits^3r_z=0 {} \,[/tex]r³ cos²Ф dz dr dФ
= [tex]\int\limits^2\pi _0 {} \, \int\limits^2_0 {} \,[/tex] 3[tex]r^{4}[/tex]cos²Ф dr dФ
= 3 [tex]\int\limits^2\pi _0 {} \,[/tex] 1/2 [tex]2^{5}[/tex]/ 5 ( 1 + cos² Ф )dr dФ
= 48 / 5 [( Ф + 1/2 sin 2Ф)[tex]]^{2\pi } _{0}[/tex]
= 96π / 5
Therefore the value of the expression is 96π/5.
To know more about the integration refer to the link given below:
https://brainly.com/question/27419605
#SPJ4
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