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anneflys

  • 2 years ago

Find the area of the region bounded by the curves y=x^2 and y=x^4

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  1. RolyPoly
    • 2 years ago
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    1. find the intercept of the curve \[x^4 - x^2 =0\]\[x^2 (x^2-1)=0\]\[x=0, 1, -1\] Then integrate it \[\int_{-1}^{0}(x^2-x^4)dx + \int_0^1(x^2-x^4)dx\]

  2. RolyPoly
    • 2 years ago
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    Or... since it is an even function, \[\int_{-1}^{0}(x^2-x^4)dx + \int_0^1(x^2-x^4)dx\]\[=2\int_{-1}^{0}(x^2-x^4)dx\]

  3. anneflys
    • 2 years ago
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    Wait, so why did you multiply everything by 2?

  4. anneflys
    • 2 years ago
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    ?

  5. RolyPoly
    • 2 years ago
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    I didn't multiply everything by two... It's just \[\int_{-1}^{0}(x^2-x^4)dx = \int_0^1(x^2-x^4)dx\] (Since it is an even function) So, \[\int_{-1}^{0}(x^2-x^4)dx + \int_0^1(x^2-x^4)dx\]\[=\int_{-1}^{0}(x^2-x^4)dx +\int_{-1}^{0}(x^2-x^4)dx \]\[=2\int_{-1}^{0}(x^2-x^4)dx \]Using the property of even function can save some work.

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