Volumes Under Surfaces & Double Integrals

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How do we findthevolumeofsuchsolids Z
we can employ a similar
riemannsum approach to find the volume
If f is a function of two
variables and was continuous y LD y
and non negative over R then the volume R
of the solid between f and region Ris
Sf f x y d A
also if f is continuous over R whose boundaryis
nottoocomplicated then the double integralexists
Corollary if the boundary of R is a smooth curve or a
finite collection ofsuch curves then the doubleintegralexists
Let f and g be functions defined on R andlet c
be a constant and if bothdoubleintegrals exist
1 SS c f x y dA CSS f x y dA
R R
2 S f x y Ig x y dA fix y dA I g x y dA
3 if R is composedof R and Ra then
If 1 t
I

4
you can integrate SiSab f x y dx dythe following
by using iterated integrals fed fab fix y dx dy
corollary this is the same as fab Sed f x y dy dx
we can write this more simply as fabfed fix y dydx
the doubleintegral can give us the volume under a surface