Triple integral

E is the solid bounded by the paraboloid  $y=x^2+z^2$ and  the plane  $y=4$  . Which of the following region is the projection of E on xy-plane ?

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A solid S is bounded by the planes  $x+2y+z=2$  , $x=2y$  , $x=0$  and  $z=0$  . View the solid S from xy-plane. Which of the following is NOT TRUE?

• Upper surface:  $z=2-x-2y$  
  Lower surface:  $z=0$  

• Upper surface:  $z=2-x-2y$  
  Interval for y:  $\left[0,\frac{x}{2}\right]$  

• Interval for x:  $\left[0,1\right]$  
  Interval for y:  $\left[\frac{x}{2},1-\frac{x}{2}\right]$  

What  $\int_{ }^{ }\int_{ }^{ }\int_{ }^{ }\ 1\ dV$  stands for?

• Area of region R

• Volume of the solid

• Mass of the solid

• Height of the solid

 $\int_0^2\int_0^{z^2}\int_0^{\left(y-z\right)}\left(2x-y\right)dxdydz$  =

•  $\frac{14}{15}$  

•  $1$  

•  $\frac{13}{15}$  

•  $\frac{16}{15}$  

The cone shape of a diamond earring has height = 2 cm and is placed at the origin of xyz axes as shown in the figure. If the density  $f\left(x,y,z\right)=x^2+y^2$  , set up the triple integral to find the mass of the cone.

•  $\int_{-2}^2\int_0^1\int_0^1\left(x^2+y^2\right)dzdydx$  

•  $\int_0^{2\cdot\pi}\int_0^2\int_r^2r^3dzdrd\theta$  

•  $\int_0^{2\cdot\pi}\int_0^2\int_{r^2}^2r^2\ dzdrd\theta$  

•  $\int_{-\pi}^{\pi}\int_0^2\int_r^4r^3\ dzdrd\theta$