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How to Find Area Between Two Curves

Finding the area between two curves is a standard problem in first-year calculus courses. It combines several mathematical techniques, including combination of functions, intersection of functions, and integration. There are a number of applications that take the form of this type of problem. For example, vector addition in physics allows the combination of various velocity functions into a single resultant function. If you wanted to determine the displacement of an object that is moving relative to a major gravitational source, say a rocket altering its orbit around the moon, you would find the area between the curves of the rocket and the moon, both relative to the same frame of reference.

Scientific or programmable calculator
Table of common integrals

Instructions

- 1
  Determine the range of values for which you are finding the area between the two curves. Either this will be given explicitly or represented as a labeled area of a graph. In the latter case, you must determine the x-values that define this bounded area. Calculate the points of intersection of the two functions.
- 2
  Within the domain defined by the x-values you calculated or were given in the previous step, determine which of the two functions is "on top" (has a greater y-value) in the bounded area of the graph.
- 3
  Create a new function by the subtraction of the "top" function from the "bottom" function.
- 4
  Determine the area under this new function by taking the definite integral between the x-values defining the bounded area of the graph (as determined in Step one).
- 5
  Solve the integral using a technique appropriate to the function, or by referring to your table of common integrals. You have now determined the area between the curves.

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