(x) \( \frac{y}{x} \frac{d y}{d x}=\frac{e^{x}}{\ln y} \)

A region is defined by the functions \( y = e^{-x} \) for \( x \geq 0 \) and \( y = 0 \). What is the volume generated when this region is revolved around the line \( y = 1 \)?

Вариант № 1 12. Найдите наименьшее значение функции \( y=(x-4) e^{x-3} \) на отрез- ке \( [2 ; 4] \).

Calculate the volume obtained by revolving the area between \( y = \frac{1}{x} \), \( y = 0 \), and \( x = 1 \) to \( x = 2 \) around the y-axis using the disc method.

\( y = \sin x \cdot \ln x \quad f ^ { \prime } ( x ) = ? \)

5. \( g(x)=\frac{x^{2}}{e^{x}} \)