Sketch the region of integrationRbounded by the paraboloidy=x2, the planey+z= 1and the xy-plane. Write the integral∫∫∫Rf(x, y, z)dV as an iterated integral in each of the three orders (a)dx dy dz, (b)dz dy dx and (c)dy dz dx. Explain your answers!
Sketch the region of integrationRbounded by the paraboloidy=x2, the planey+z= 1and the xy-plane. Write the
We are given the following surfaces:
paraboloid:
plane 1: y + z = 1
plane 2: z = 0 (XY-plane)
We have to express the given bounded region in the form iterated integral.
The given surfaces can be plotted as
1. dx dy dz
First, we find the limits of x.
The projection of the solid on the XY plane can be plotted as 
Thus, the range of x is to as .
Now, from the following figure we can estimate the range of y.
The range of y becomes 0 to 1-z.
The range of z becomes 0 to 1.
Therefore, the iterated integral becomes,
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