A solid is formed with triangular cross-sections where each triangle's base is defined by the equation \(y = x^2\) on interval \([0, 2]\). What is the volume of this solid?

Sea \[ f(x)=\left\{\begin{array}{ll}\left(\frac{\sin (2 x)}{x}\right) e^{-x^{2}} & \text { si } x \neq 0 \\ 2 & \text { si } x=0\end{array}\right. \] (a) Demostrar que \( f(x) \) es continua en \( x=0 \). (b) Demostrar que \( f(x) \) es derivable en \( x=0 \) y su derivada es continua en \( x=0 \).

\( L ( e ^ { 3 t } \sinh 4 t ) \)

\( \lim _ { x \rightarrow 0 } ( \frac { t 9 ( x ) } { x } ) ^ { \frac { 1 } { x ^ { 2 } } } \)

Calcular los límites cuando \( x \) tiende a cero de las funciones: (a) \( \frac{\cos x-\frac{x^{2}}{2}-1}{x^{3}} \) (b) \( \frac{\sin x+e^{x}-1}{x} \) (c) \( \frac{e^{x}-1}{x} \)

A car's velocity is given by the function \( v(t) = 5t^2 + 3 \). Calculate the distance traveled by the car from time \( t = 0 \) to \( t = 4 \).