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Find the area of the region between the two curves and
Foundations:
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1. You can find the intersection points of two functions, say
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by setting and solving for
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2. The area between two functions, and is given by
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for where is the upper function and is the lower function.
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Solution:
Step 1:
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First, we need to find the intersection points of these two curves.
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To do this, we set
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Getting all the terms on one side of the equation, we get
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Therefore, we get that these two curves intersect at
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Hence, the region we are interested in occurs between and
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Step 2:
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Since the curves intersect also intersect at this breaks our region up into two parts,
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which correspond to the intervals and
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Now, in each of the regions we need to determine which curve has the higher value.
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To figure this out, we use test points in each interval.
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For we have
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and
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For we have
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and
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Hence, the area of the region bounded by these two curves is given by
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Step 3:
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Now, we integrate to get
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Final Answer:
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