Triple Integrals. z-Simple, y-simple, z-simple Approach. z-Simple solids (Type 1). Definition: A solid region E is said to be z-Simple if it is bounded by two surfaces z=z 1 (x,y) and z=z 2 (x,y) (z 1 £ z £ z 2 ). Iterated Triple Integrals over z-Simple solid E.
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This gives the volume V over the region Dxy in the xy-plane of the surface z=z2(x,y)
used so that dAxz can be written as
r dr dq instead ofdx dz.
Compare to a cylinder of radius and height 16 which has
double this volume (anyone know why?) and contains our solid E inside it.
Often a solid is simple in more than one variable.An alternate approach is to look for the one variable that it is not simple in, and make that the outer limitof integration. The inner limit is then a double integral.
This approach is also helpful in sketching the solid of integration, because as we will see the outer limit ofintegration corresponds to constant values on whichcontour regions in the simple plane lie
where Dxy(z) is the trace of the solid (a trace of a solid is a region instead of a curve) in the plane z=constant.