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The value of the integral is [tex]\int\limits^{}_s {9x + 2y + z^2} \, dS = \frac{4}{3}\pi[/tex]
The expression is given as:
[tex]\int\limits^{}_s {9x + 2y + z^2} \, dS[/tex]
[tex]x^2 + y^2 + z^2 = 1[/tex]
Rewrite the integral as:
[tex]\int\limits^{}_s {9x + 2y + z*z} \, dS[/tex]
As a general rule, we have:
[tex]\int\limits^{}_s {Px + Qy + R*z} \, dS[/tex]
By comparison, we have:
P = 9
Q = 2
R = z
By the divergence theorem, we have:
F = Pi + Qj + Rk
So, we have:
F = 9i + 2j + zk
Differentiate
F' = 0 + 0 + 1
F' = 1
The volume of a sphere is:
[tex]V = \frac{4}{3}\pi r^3[/tex]
Where:
r = F' = 1
So, we have:
[tex]V = \frac{4}{3}\pi (1)^3[/tex]
Evaluate
[tex]V = \frac{4}{3}\pi[/tex]
This means that:
[tex]\int\limits^{}_s {9x + 2y + z^2} \, dS = \frac{4}{3}\pi[/tex]
Hence, the value of the integral is [tex]\int\limits^{}_s {9x + 2y + z^2} \, dS = \frac{4}{3}\pi[/tex]
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