Find the upper and lower boundary curve to find the area between two curves.

  • Thread starter theBEAST
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How do I know which function is the upper boundary curve and which is the lower boundary curve. For example find the area between the curves e^x and x bounded on the sides x=0 and x=1. We can draw it and we know that e^x is the upper curve and x is the lower curve. Thus the area is ∫e^x-∫x. However, let's say I did not know how to draw the function, I could easily make the mistake and and solve Area = ∫x-∫e^x. The answer would be negative. Or say we had two very complicated functions, how would I know which is the upper and which is the lower boundary curve?
 

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  • #2
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If you had two functions, and wanted to know which one was "above" the other at any given point, you would just plug in the point and see which function returned a higher value. Ex: e^1 > 1, so e^x > x at x = 1.

When you graph a function you are essentially calculating the value of the function at every single point in the domain.

If you have complicated functions which are difficult to graph, you could determine intervals where the graphs of the functions crossed each other, and then test values in each of those intervals to see which function is dominant in each interval.

Using the above functions, you could set e^x = x, then solve for x to find where the two functions meet (if at all). Then use that value to separate your domain into intervals. Then test values in each interval to determine which function is above the other.
 
  • #3
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you can take the derivative which gives you the rate of change. Then, if you want to integrate from say starting from the origin, evaluate the derivatives at zero, and the greater of the two would be on top. If you have something like f=x^2 & g=x^4, then f'=2x & g'=4x^3. At the origin, they both are zero, so consider a small perturbation. f'(0.1)=.2 and g'(0.1)=.004, so f is above g close to the origin. You then have to consider if the two curves intersect. f=g at x=1 and x=-1. So again, evaluate f' and g' at x=1. So f'=2 and g'=4, so now g is on top of f for x>1. Similarly, you can see what happens at x=-1. Hope that helps.
 

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