la curva \( y=x^{2} \) se gira alrededor del eje \( x \) desde \( x=0 \) hasta \( x=2 \). calcular el volumen del sólido resultante utilizando el método de discos

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the \( x \)-axis. \( y=3 x, y=6, x=0 \) \( V=\square \) (Type an exact answer, using \( \pi \) as needed.)

3) Sean las funciones: \[ f(x)=e^{x^{2} \cdot e^{2 x}} \quad g(x)=\left(\frac{x-3}{x^{2}+1}\right)^{2} \] Determine: a) \( f^{\prime}(x) \) b) \( g^{\prime}(x) \) c) Calcule el valor de \( \frac{g^{\prime}(0)-3 f(0)}{g(0)} \)

4) Considere la función \( f(x)=x^{3}+2 x^{2}-3 x-1 \) a) Determine \( f^{\prime}(x), f^{\prime \prime}(x) \) y \( f^{\prime \prime \prime}(x) \) b) Encuentre el valor de \( x \) en la ecuación \( 2 f^{\prime}(x)-x f^{\prime \prime}(x)=f^{\prime \prime \prime}(x) \)

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the \( x \)-axis. \( y=\sqrt{x}, y=1, x=0 \) \( V=\square \) (Type an exact answer, using \( \pi \) as needed.)

Use Euler's method to find an approximation for \( y(0.5) \) in the initial value problem \( y' = \frac{dy}{dx} = 4y + x \) with \( y(0) = 1 \) and \( h = 0.1 \).