Question

Use the Divergence Theorem to evaluate

Answer 1

The value of the integral is $\int\limits^{}_s {9x + 2y + z^2} \, dS = \frac{4}{3}\pi$

How to evaluate the integral?

The expression is given as:

$\int\limits^{}_s {9x + 2y + z^2} \, dS$

$x^2 + y^2 + z^2 = 1$

Rewrite the integral as:

$\int\limits^{}_s {9x + 2y + z*z} \, dS$

As a general rule, we have:

$\int\limits^{}_s {Px + Qy + R*z} \, dS$

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:

$V = \frac{4}{3}\pi r^3$

Where:

r = F' = 1

So, we have:

$V = \frac{4}{3}\pi (1)^3$

Evaluate

$V = \frac{4}{3}\pi$

This means that:

$\int\limits^{}_s {9x + 2y + z^2} \, dS = \frac{4}{3}\pi$

Hence, the value of the integral is $\int\limits^{}_s {9x + 2y + z^2} \, dS = \frac{4}{3}\pi$

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