A nice image is to imagine a small clementine, and a large orange, with the same thickness of peel.

Peel each of them (a reduction of "dr" in the radius). Look at the pile of peel. Which is bigger? Why? The peel you take off is the reduction in volume!

At the bottom of this example (and many others) is the observation that the rate of change of volume with respect to radius is the surface area. Which has the larger surface area (total peel)?

Walter Whiteley