In this video I explain how to find the volume under a paraboloid and a disk using double integration and polar coordinates. I begin by defining the region of the disk and converting it from cartesian coordinates to polar coordinates.
Once the region is defined I demonstrate how we can sum infinitesimally small pieces of area in the r direction to generate an infinitesimally small slice of area which is defined by the inner integral. I then explain how the small slices of area can be summed by rotating about the angles of Θ, and show that the outer integral is evaluated between angles of Θ between 0 and π.
Having defined the double integrals associated with the region I demonstrate how, when the area is multiplied by a height function it generates the volume of the solid between the paraboloid and disk.
I then move on to evaluating the integrals beginning with the inner integral followed by the outer integral to find the volume.