Area Between Curves

Calculus 1 · Axiom Academy

LESSON Area Between Two Curves When two curves trap a region, the integral of the gap between them measures its area. Take an upper curve y = f(x) and a lower curve y = g(x) that cross at two points. Between those crossings, f stays above g , and the region they enclose is shaded below. At any x in the interval, a single vertical segment reaches from the bottom edge up to the top edge — its length is the gap f(x) - g(x) . the height of the region at each x is the top curve minus the bottom curve 2. Where They Meet, and Who Is on Top The gap only makes sense over the interval where the region actually exists — from one crossing to the next. Those crossings are the limits of integration . To find them, set the curves equal and solve. Then, to know which function is the top and which is the bottom, test a single point between the crossings: whichever value is larger is the upper curve there. Then f is the top across [a,b] , and the height is f(x) - g(x) . Then g is on top instead — just swap the order so the height stays positive. 3. Slice the Gap, Then Add It Up Picture the region as a stack of thin vertical strips. Each strip sits at some x , has width dx , and reaches the full height of the gap, so its area is . Sweeping across the interval and accumulating every strip turns the sum into a definite integral — the running total below is the area gathered so far.

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