Example 1. Determine the area bounded by the lines $x y=6$, $x=1, x=e, y=0$ (Fig. 5.5).

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-208.jpg?height=427&width=347&top_left_y=1144&top_left_x=56)

Fig. 5.5

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-208.jpg?height=423&width=480&top_left_y=1141&top_left_x=582)

Fig. 5.6
Solution. Defining $y$ from the hyperbola equation $x y=6$, we get $y=\frac{6}{x}$. From the condition, it follows that $a=1, b=e$. Substituting the values of $a, b$ and $\frac{6}{x}$ (the expression for $y$) into formula (5.13), we find:

$$
S=\int_{1}^{e} \frac{6}{x} d x=6 \int_{1}^{e} \frac{d x}{x}=\left.6 \ln x\right|_{1} ^{e}=6(\ln e-\ln 1)=6(1-0)=6
$$