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Understanding Areas Between Curves and Graph Behavior Relative to the X-axis

This text explores the calculation of areas between curves represented by functions f(x) and g(x), particularly when one or both graphs may fall below the x-axis. It discusses how to identify points of intersection, determine which function is above or below, and compute the integral for the area correctly. The key takeaway is that even if functions cross the x-axis, the integral still provides a valid geometric area between the curves, as long as we are mindful of the signs in our calculations.

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Understanding Areas Between Curves and Graph Behavior Relative to the X-axis

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  1. g f Areas Between Curves a b

  2. g a b f What if the green graph falls below the x-axis somewhere? Do the arrows match the equation we came up with before?

  3. a b g f What if both curves lie partially below the x-axis?

  4. a b g f What if both curves lie entirely below the x-axis?

  5. In Summary If the graph of the function f lies above the graph of g on the interval [a,b], the integral gives us actual geometric area between the graphs of g and f, whether or not one or both of the functions drops below the x-axis.

  6. So what do we do? • We find the points of intersection of the graphs of f and g. That is, we solve the equation f (x)=g(x) for x. • We determine which of the two functions is above and which is below. (How might we do this?) • Then we compute the integral.

  7. True or false We don’t really have to determine which function is above and which is below. We can just compute and take the absolute value of the result. (Since the answer will obviously just be “off” by a minus sign if we subtract the wrong function.)

  8. g g f b a c a b f Well, sort of, but you have to be Cautious. . . Works here! Does not work here!

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