How to solve double integrals

From the course of mathematical analysis known concept of double integral. Geometrically, the double integral represents the volume of the cylindrical body based on D and bounded by the surface z = f(x, y). Using double integrals to calculate the mass of a thin plate with a given density, the area of a plane figure, the area of the piece of surface, the coordinates of the center of gravity of a homogeneous plate and the other quantities.

Instruction

1. The solution of the double integrals can be reduced to the computation of definite integrals.
   If the function f(x, y) is a closed and continuous in a region D bounded by the line y = c and the line x = d, with c < d and a function y = g(x) and y = z(x), with g(x), z(x) is continuous on [c; d] and g(x) ? z(x) on this interval, then calculate the double integral by the formula presented in the figure.
   
2. If the function f(x, y) is a closed and continuous in a region D bounded by the line y = c and the line x = d, with c < d and a function y = g(x) and y = z(x), with g(x), z(x) is continuous on [c; d] and g(x) = z(x) on this interval, then calculate the double integral by the formula presented in the figure.
   
3. If you want to calculate the double integral more complex regions D, the region D is split into parts, each of which represents an area described in paragraph 1 or 2. Calculates the integral for each of these areas, the results obtained are summarized.