While extending the definition of Riemann integral from the case that calculates area to the case the calculates volume is not difficult, actually calculating these limits, in 2D or 3D, is difficult. The following steps illustrate a theorem that makes calculation of these integrals easier.
Activity 53.
Consider the surface in 3D \(g(x,y)=5-3x^2-2y^2\) on the domain \(x \in [-1,1]\) \(y \in [-1,1].\)
(a)
Graph \(g(x,y)\) with \(y=-1\text{.}\)
(b)
Calculate the area between the line \(y=-1\) and this curve using an integral.
(c)
Graph \(g(x,y)\) with \(y=0\text{.}\)
(d)
Calculate the area between the line \(y=0\) and the curve \(g(x,0)\) using an integral.
(e)
What is the graph of \(g(x,y)\) with \(y=c\text{,}\) where \(c\) is a constant?
(f)
Calculate the area between the line \(y=c\) and the curve \(g(x,c)\) using an integral.
(g)
Of how many of these curves, which define an area, does the surface, which defines a volume, consist?
(h)
Graph \(g(x,y)\) with \(x=-1\text{.}\)
(i)
Calculate the area between the line \(x=-1\) and the curve \(g(-1,y)\) using an integral.
(j)
Graph \(g(x,y)\) with \(x=0\text{.}\)
(k)
Calculate the area between the line \(x=0\) and the curve \(g(0,y)\) using an integral.
(l)
What is the graph of \(g(x,y)\) with \(x=a\) where \(a\) is a constant?
(m)
Calculate the area between the line \(x=a\) and the curve \(g(a,y)\) using an integral.
(n)
Of how many of these curves, which define an area, does the surface, which defines a volume, consist?
(o)
Describe in general terms how these areas could be used to calculate the volume.
Theorem 3.3.3. Fubini.
If \(f\) is continuous on the rectangular domain \(R\) defined by \(x \in [a,b]\) and \(y \in [c,d]\) then
\begin{equation*}
\dint_R f(x,y) \; dA = \int_a^b \int_c^d f(x,y) \; dy dx = \int_c^d \int_a^b f(x,y) \; dx dy
\end{equation*}
This theorem still works if the domain is not rectangular and any discontinuities are limited to a finite number of smooth curves.
