Let $a$ be a positive real number. Let P denote the point of intersection of the following two curves

$$\begin{aligned}
& C _ { 1 } : y = \frac { 3 } { x } \\
& C _ { 2 } : y = \frac { a } { x ^ { 2 } } ,
\end{aligned}$$

and let $\ell$ denote the tangent to $C _ { 2 }$ at P. Then we are to find the area $S$ of the region bounded by $C _ { 1 }$ and $\ell$.

Since the coordinates of P are $\left( \frac { a } { \mathbf { N } } , \frac { \mathbf { O } } { a } \right)$, the equation of $\ell$ is

$$y = - \frac { \mathbf { P Q } } { a ^ { 2 } } x + \frac { \mathbf { R S } } { a }$$

When we set

$$p = \frac { a } { \mathbf { T } } , \quad q = \frac { a } { \mathbf { U } } \quad ( p < q )$$

$S$ is obtained by calculating

$$S = [ \mathbf { V } ] _ { p } ^ { q }$$

where $\mathbf{V}$ is the appropriate expression from among (0) $\sim$ (5) below.

(0) $\frac { 18 } { a ^ { 2 } } x ^ { 2 } - \frac { 27 } { a } x + 3 \log | x |$\\
(1) $\frac { 9 } { a ^ { 2 } } x ^ { 2 } - \frac { 9 } { a } x + 3 \log | x |$\\
(2) $- \frac { 27 } { a ^ { 2 } } x ^ { 2 } + \frac { 18 } { a } x - 3 \log | x |$\\
(3) $- \frac { 27 } { a ^ { 2 } } x ^ { 2 } + \frac { 27 } { a } x - 3 \log | x |$\\
(4) $\frac { 27 } { a ^ { 2 } } x ^ { 2 } - \frac { 27 } { a } x + 3 \log | x |$\\
(5) $- \frac { 18 } { a ^ { 2 } } x ^ { 2 } + \frac { 27 } { a } x - 3 \log | x |$

Hence we obtain

$$S = \frac { \mathbf { W } } { \mathbf { X } } - 3 \log \mathbf { Y }$$