Setting up Double Integrals 

Let's look at how to compute the volume of a function in 3-space bounded by a region R in the plane. 

Look at the region R bounded by the line y = x, y = 2 - x, and x = 0. 

The red and green lines will be the edges of the vertical surfaces. 

The top of the solid will be the paraboloid z = 4 - x^2 - y^2. 

Let's begin to create the object by generating the vertical portions of the figure. Here we do the vertical edge for the line y=x 

And this is the vertical edge for the line y= 2-x 

Now we put the two together... 

Here we add the surface z = 4-x^2 - y^2 

Lastly we add the final edge, x = 0 

Now that we know what our solid looks like we can perform the double integration. Here we use the int(...) function to perform the integration. 

Start by doing the integration piecemeal, first integrating with respect to y and following that by x  

Or we can perform the double integration and use evalf(...) to get a number result