Determine o volume de um sólido formado ao rotacionar uma semicircunferência de raio 3 em torno do seu diâmetro.

What is the solution to the following differential equation, subject to the initial condition \( y(0)=2 \) ? \( \frac{d y}{d x}=6 x+6 \)

A. The critical number of \( f(x) \) B. Find the intervals on which \( f \) is increasing and decreasing by using the first derivative test Find the local maximum and minimum, if they exist by using the first derivative test Find the intervals on which \( f \) is concave up and concave down Find the inflection points of \( f(x) \)

What is the general solution to the following: \( \int 2 x \sin (x) d x \) ?

Un objeto se mueve a lo largo de una línea y su aceleración está dada por \( a(t) = 12t \). Si en el tiempo inicial \( t=0 \) la velocidad es \( v(0)=4 \), encuentra la expresión para la velocidad.

Integrate the following: \( \int x e^{2 x} d x \)