Find the volume of the given solid. Enclosed by the paraboloid $z=7 x^{2}+4 y^{2}$ and the planes $x=0, y=4, y=x, z=0$

Step 1 ：The volume of the solid enclosed by the given surfaces can be found by integrating over the region defined by the surfaces. The limits of integration can be determined by the intersection points of the surfaces.

Step 2 ：The volume element in Cartesian coordinates is dx*dy*dz. However, since the region is defined by the planes \(x=0\), \(y=4\), \(y=x\), \(z=0\) and the paraboloid \(z=7 x^{2}+4 y^{2}\), it would be easier to use polar coordinates to express the volume element.

Step 3 ：In polar coordinates, the volume element is r*dr*dθ*dz. The limits of r and θ can be determined by the intersection points of the planes, and the limit of z can be determined by the paraboloid.

Step 4 ：Perform the integration to find the volume, which gives the formula: volume = 22*pi*r**2

Step 5 ：The volume is a function of r, which is the radius in polar coordinates. The volume is proportional to the square of the radius, which is consistent with the formula for the volume of a cylinder. The constant of proportionality is 22*pi, which is the result of the integration over the limits of r, theta, and z.

Step 6 ：Final Answer: The volume of the solid enclosed by the given surfaces is \(\boxed{22\pi r^{2}}\).