Get the equation of the elliptic paraboloid. This information is necessary to calculate the volume. For example purposes, assume the equation is z = x^2 + 4y^2.
Find the coordinates of the square or the rectangle. This information is also required because you are calculating the volume between an elliptic paraboloid and a square or rectangular surface. For example purposes, assume that the surface is a square and the coordinates are (-1, 1) for the x-plane and (-2, 2) for the y-plane. The term "plane" rather than "axis" is used to refer to a three-dimensional surface.
Calculate the volume under the elliptic paraboloid and over the square or rectangular surface. A double integration is required -- first integrate along the x-plane, then along the y-plane. In the example, integrating along the x-plane results in the equation x^3/3 + 4xy^2. Integrate this equation from x = -1 to x = 1 to get 1/3 + 4(1)y^2 - [-1/3 + 4(-1)y^2] = 1/3 + 4y^2 + 1/3 + 4y^2 = 2/3 + 8y^2. Integrate this equation along the y-plane results in (2/3)y + (8/3)y^3. Integrate this equation from y = -2 to y = 2 to get 4/3 + 64/3 - (-4/3 - 64/3) = 136/3. Therefore, the volume under the elliptic paraboloid and over the square surface is 136/3, or roughly 45.33 units.