\[f(x,y,z)\]
over a region of space defined by the curves \[g(x,y)=z, \; h(x)=y, a \lt x \lt b\]
.\[\int_V f(x,y,z)dV= \int^b_a \int^{Max(h(x))}_{Min(h(x))} \int^{Max(g(x,y))}_{Min(g(x,y))} f(x,y,z) dzdydx\]
.Example: A function
\[f(x,y,z)\]
is to be evaluated over the volume between the paraboloid \[x^2+y^2=z, \; x^2+y^2=4\]
and the \[xy\]
plane. Analysis of these equations gives \[0 \le z \le x^2+y^2, \; - \sqrt{4-x^2} y \le \sqrt{4-x^2}, \; -2 \le x \le 2\]
.The integral becomes
\[\int^2_{-2} \int^{\sqrt{4-x^2}}_{- \sqrt{4-x^2}} \int^{x^2+y^2}_{0} f(x,y,z) dzdydx\]
The process is to find one of the equations which gives one of the variables in terms of the others successively, finding the limits for that variable in terms of the other variables for each integral successively reducing the number of variables by 1.