If under pure competition demand and supply functions are given by $p = \sqrt{10 - x}$ and $p = \frac{1}{2}(x-2)$ respectively, where $p$ is price per unit and $x$ is quantity, then the consumer surplus is:

To find the Consumer Surplus (CS), we first need to determine the equilibrium point $(x_0, p_0)$ where the demand and supply curves intersect.

Set the demand function equal to the supply function:

$\sqrt{10 - x} = \frac{1}{2}(x - 2)$

Square both sides to solve for $x$:

$10 - x = \frac{1}{4}(x - 2)^2$

$40 - 4x = x^2 - 4x + 4$

$x^2 = 36 \Rightarrow \mathbf{x_0 = 6}$

(We take the positive root as quantity cannot be negative).

Now, find the equilibrium price $p_0$ by substituting $x_0$ into the supply equation:

$p_0 = \frac{1}{2}(6 - 2) = \mathbf{2}$

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The formula for Consumer Surplus is:

$CS = \int_{0}^{x_0} f(x) \, dx - (p_0 \cdot x_0)$

Where $f(x) = \sqrt{10 - x}$.

$CS = \int_{0}^{6} \sqrt{10 - x} \, dx - (2 \cdot 6)$

Let $u = 10 - x$, then $du = -dx$.

$\int \sqrt{10 - x} \, dx = -\frac{(10 - x)^{3/2}}{3/2} = -\frac{2}{3}(10 - x)^{3/2}$

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Evaluate from 0 to 6:

$\left[ -\frac{2}{3}(10 - 6)^{3/2} \right] - \left[ -\frac{2}{3}(10 - 0)^{3/2} \right]$

$= -\frac{2}{3}(4)^{3/2} + \frac{2}{3}(10)^{3/2}$

$= -\frac{2}{3}(8) + \frac{2}{3}(10\sqrt{10}) = \frac{20\sqrt{10} - 16}{3}$

Subtract the rectangle area ($p_0 \cdot x_0 = 12$):

$CS = \frac{20\sqrt{10} - 16}{3} - 12$

$CS = \frac{20\sqrt{10} - 16 - 36}{3}$

$CS = \frac{20\sqrt{10} - 52}{3}$

Factor out a 2:

$CS = \boxed{\frac{2}{3}[10\sqrt{10} - 26]}$