Area Between Curves

Area Between Curves - Area Between Curves In this section...

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Area Between Curves In this section we are going to look at finding the area between two curves. There are actually two cases that we are going to be looking at. In the first case we are want to determine the area between and on the interval [ a,b ]. We are also going to assume that . Take a look at the following sketch to get an idea of what we’re initially going to look at. In the Area and Volume Formulas section of the Extras chapter we derived the following formula for the area in this case. (1) The second case is almost identical to the first case. Here we are going to determine the area between and on the interval [ c,d ] with .
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In this case the formula is, (2) Now (1) and (2) are perfectly serviceable formulas, however, it is sometimes easy to forget that these always require the first function to be the larger of the two functions. So, instead of these formulas we will instead use the following “word” formulas to make sure that we remember that the formulas area always the “larger” function minus the “smaller” function. In the first case we will use, (3) In the second case we will use,
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(4) Using these formulas will always force us to think about what is going on with each problem and to make sure that we’ve got the correct order of functions when we go to use the formula. Let’s work an example. Example 1 Determine the area of the region enclosed by and . Solution First of all, just what do we mean by “area enclosed by”. This means that the region we’re interested in must have one of the two curves on every boundary of the region. So, here is a graph of the two functions with the enclosed region shaded. Note that we don’t take any part of the region to the right of the intersection point of these two graphs. In this region there is no boundary on the right side and so is not part of the enclosed area. Remember that one of the given functions must be on the each boundary of the enclosed region. Also from this graph it’s clear that the upper function will be dependent on the range of x ’s that we use. Because of this you should always sketch of a graph of the region. Without a sketch it’s
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often easy to mistake which of the two functions is the larger. In this case most would probably say that is the upper function and they would be right for the vast majority of the x ’s. However, in this case it is the lower of the two functions.
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.

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Area Between Curves - Area Between Curves In this section...

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