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 \( y = 1 \)?

\( \left. \begin{array} { l } { \frac { d } { d x } ( \sin ^ { 2 } x ^ { 2 } ) } \\ { \frac { d } { d x } ( \sin x ) ^ { 2 } } \\ { = 2 \cos x } \end{array} \right. \)

\( \left. \begin{array} { l }{ \frac { d } { d x } ( x ^ { 2 } \sin ^ { 2 } x ^ { 2 } ) } \\ { = x ^ { 2 } \frac { d } { d x } ( \sin ^ { 2 } x ^ { 2 } ) + \sin ^ { 2 } x ^ { 2 } \frac { d } { d x } ( x ^ { 2 } ) } \\ { = x ^ { 2 } \frac { d } { d x } ( \sin x ) ^ { 2 } + \sin ^ { 2 } x ^ { 2 } \frac { d } { d x } ( x ^ { 2 } ) } \\ { = x ^ { 2 } ( 2 \cos x ) + \sin ^ { 2 } x ^ { 2 } 2 x } \\ { = 2 x ^ { 2 } \cos x + 2 x \sin ^ { 2 } x ^ { 2 } } \end{array} \right. \)

4. (15б) Выясните, является ли функция \( z=\ln \left(x^{2}+y^{2}+2 y+1\right) \) решением уравнения \( \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 \)

(vi) \( \mathrm{L}^{-1}\left\{\tan ^{-1} \frac{1}{\mathrm{p}}\right\}= \)

a) \( D[(1-x) \sqrt{1+x}] \)