Calcule o volume de um sólido gerado pela rotação da região delimitada pela curva \( y = x^{2} \) e a linha \( y = 4 \) em torno do eixo x, onde as seções transversais são triângulos isósceles com base na linha do limite.

Evaluate the triple integral. \[ \iiint_{G} x y z d V \] where \( G \) is the solid in the first octant that is bounded by the parabolic cylinder \( z=3-x^{2} \) and the planes \( z=0, y=x \) and \( y=0 \).

\( \int \frac { \sqrt { x ^ { p } - \gamma \Delta } } { x } d x = \)

Evaluate the following triple integral in spherical coordinates. \[ f(\rho, \theta, \varphi)=\sin (\varphi) \] over the region \( 0 \leq \theta \leq 2 \pi, 0 \leq \varphi \leq \frac{\pi}{4}, \quad 7 \leq \rho \leq 8 \)

Evaluate the triple integral. \[ \iiint_{G} x y \sin (y z) d V \] where \( G \) is the rectangular box defined by the inequalities \( 0 \leq x \leq \pi, 0 \leq y \leq 17,0 \leq z \leq \frac{\pi}{102} \)

\( \int _{0}^{1}\frac{1-\cos (x)}{\sqrt {x^{5}}+5x^{3}} d x \)