Calc07_2 - 7.2 Areas in the Plane Gateway Arch St Louis...

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7.2 Areas in the Plane Gateway Arch, St. Louis, Missouri Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003
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2 1 2 y x = - 2 y x = - How can we find the area between these two curves? We could split the area into several sections, use subtraction and figure it out, but there is an easier way.
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2 1 2 y x = - 2 y x = - Consider a very thin vertical strip. The length of the strip is: 1 2 y y - or ( 29 ( 29 2 2 x x - - - Since the width of the strip is a very small change in x , we could call it dx .
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2 1 2 y x = - 2 y x = - 1 y 2 y 1 2 y y - dx Since the strip is a long thin rectangle, the area of the strip is: ( 29 2 length width 2 x x dx = - + If we add all the strips, we get: 2 2 1 2 x x dx - - +
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2 1 2 y x = - 2 y x = - 2 2 1 2 x x dx - - + 2 3 2 1 1 1 2 3 2 x x x - - + 8 1 1 4 2 2 3 3 2 - + - - + + 8 1 1 6 2 3 3 2 - + - - 36 16 12 2 3 6 - + - - 27 6 = 9 2 =
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The formula for the area between curves is: ( 29 ( 29 1 2 Area b a f x f x dx = - We will use this so much, that you won’t need to “memorize” the formula!
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