My calculus teacher in high school, Mr. Joffray, had a memorable metaphor for this fluid scenario, with its sliding x and its changing area. He asked us to imagine a magical paint roller moving sideways. As it rolls steadily to the right, it paints the region under the curve gray. The dotted line at x marks the current position of this imaginary roller as it rolls to the right. Meanwhile, to ensure that the region is painted neatly, the roller instantly and magically shrinks or stretches in the vertical direction,
