Volume Bounded by a Surface and a Region in the Plane

Suppose we have a surface  
  is the height of the surface above the  

The volume bounded by the surface and the region  
  in the  
  plane is  
\[V = \int_R f(x,y) dx dy\]
Example: Find the volume bounded by the surface  
\[z=x^3 + y^3\]
  and the triangle in the  
  plane between the coordinates  
\[(0,0,0), ((1,1,0), (1,0,0)\]

The triangle has boundaries  
\[x=0, x=1, y=x\]
  so we can write as the region of integration as  
\[0 \leq x \leq 1 , y \leq x\]

The integral becomes
\[\begin{equation} \begin{aligned} V &= \int^1_0 \int^x_0 (x^3 +y^3) dy dx \\ &= \int^1_0 [x^3 y + y^4/4]^x_0 dx \\ &= \int^1_0 5x^4/4 dx \\ &= [x^5/4]^1_0 \\ &= 1/4 \end{aligned} \end{equation} \]

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