Shaded Regions Between Curves

Calculus 1 · Axiom Academy

The area trapped between two curves is just one integral — of the top curve minus the bottom one. Two curves, one region — and a single integral that measures it You already know the integral measures the area under one curve. But a lot of the areas you actually want — the petal between two parabolas, the sliver between a line and a curve — are trapped between two graphs. The surprise is that you don't need anything new: the same integral handles it, once you aim it at the right quantity. Watch the sweep line cross the region from left to right. Everything it passes between the two curves fills in, and the running total climbs to the exact area trapped between them. The area between the curves is just accumulated gap — the running total the sweep line leaves behind, settling at . Where does the region begin and end? The region isn't infinite — it's pinched shut wherever the two curves meet. Drag the two bounds (or the handles on the plot) until they land exactly where the curves cross. The shaded region grows and shrinks to match. You're solving f(x) = g(x) by eye. Where f - g = 0 , the curves touch and the region closes — those crossings are the integration bounds. The height of each slice is top minus bottom

This is the written version of the interactive lesson above. See the full Calculus 1 course.