A classical mechanics problem by Rocco Dalto

Let $B:(x,y,z)$ be any point not on the given line and choose the points $0:(x_{0},y_{0},z_{0})$ and $A:(x_0 + a,y_{0} + b,z_{0} + c)$ on the given line. $$

The distance from point $B$ to the given line is $d = \dfrac{|\vec{OB} \: X \: \vec{OA}|}{|\vec{OA}|}$
$$

$\vec{0A} = a \vec{i} + b \vec{j} + c \vec{k}$ $$

$\vec{0B} = (x - x_{0}) \vec{i} + (y - y_{0}) \vec{j} + (z - z_{0}) \vec{k}$ $$

$\vec{OB} \: X \: \vec{OA} = (c(y - y_{0}) - b(z - z_{0})) \vec{i} + (a(z - z_{0}) - c(x - x_{0})) \vec{j} + (b(x - x_{0}) - a(y - y_{0})) \vec{k}$ $$

$$

$\implies d^2 = \dfrac{(c(y - y_{0}) - b(z - z_{0}))^2 + (a(z - z_{0}) - c(x - x_{0}))^2 + (b(x - x_{0}) - a(y - y_{0}))^2}{a^2 + b^2 + c^2}$ $$

$\therefore I = \int_{S} \int d^2 \: dA =$ $$

$\int_{S} \int \dfrac{(c(y - y_{0}) - b(z - z_{0}))^2 + (a(z - z_{0}) - c(x - x_{0}))^2 + (b(x - x_{0}) - a(y - y_{0}))^2}{a^2 + b^2 + c^2} \: dA$