\( \left. \begin{array} { l } { x _ { c m } = \frac { 1 } { A } \int \int x d x d y } \\ { A = a ^ { 2 } \quad x = b \quad y = a } \end{array} \right. \)

Resuelva las Integrales: a) \( \left.\int\left(3 x^{1 / 2}+7\right) d x, b\right) \int \frac{2 x+1}{\sqrt{x^{2}}+1} d x, \int_{0}^{2}\left(3 x^{2}+2 x\right) d x \) Valor (30 puntos)

Graph the polynomial function \( f(x)=-2(x-2)^{2}\left(x^{2}-9\right) \) using parts (a) through (e). (a) Determine the end behavior of the graph of the function. The graph of f behaves like \( y=\square \) for large values of \( |x| \). (b) Find the \( x \) - and \( y \)-intercepts of the graph of the function. (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.) The \( y \)-intercept is \( \square \). (Simplify your answer. Type an integer or a fraction.) (c) Determine the real zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the \( x \)-axis at each \( x \)-intercept. The real zero(s) of \( f \) is/are \( \square \).

\( \int \frac { d x } { 4 x ^ { 2 } + 25 } \)

Find the general indefinite integral. (Use \( C \) for the constant of integration. \[ \int 5 v\left(v^{2}+2\right)^{2} d v \]

La región acotada por \( 2.5 x^{2} \) y \( 3 x \) se hace girar con respecto al eje \( y \), formando asi el sólido de revolución. Si utilizáramos la técnica de cascarones cilindricos, cCual de las siguientes expresiones sería la correcta para calcular el volumen? Tome en cuenta que el volumen resultante es de 27 unidades cúbicas.