Find the volume of the solid formed by rotating the area under the curve \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 3 \) about the y-axis.

Hallar y' de \( y=4 x^{7} \)

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.

- Derivative of \( \tan x \) Solution

On considère l'équation différentielle \[ (E): y^{\prime \prime}-\frac{1}{1+t^{2}} y^{\prime}+y=0 \] On admet le théorème de Cauchy-Lipschitz Pour tout \( \left(t_{0}, y_{0}, y_{0}^{\prime}\right) \in \mathbb{R}^{3} \), il existe une unique solution de \( (E) \) vérifiant : \( y\left(t_{0}\right)=y_{0} \) et \( y^{\prime}\left(t_{0}\right)=y_{0}^{\prime} \) Question : Justifier que si \( y \) est une solution de \( (E) \) vérifiant \( y\left(t_{0}\right)=y^{\prime}\left(t_{0}\right)=0 \), alors \( y \) est nulle Partie I : Théorème de Relèvement anominam

\( f ( x ) = x ^ { 3 } - 4 x ^ { 2 } \rightarrow f ( 0 ) , f ( 1 ) , f ( - 1 ) , f ( \frac { 3 } { 2 } ) \)