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    The diagram shows the graph of, where a ∈ ] 1, ∞ [.
    The area of the pink region is equal to the area of the blue region. Give two equations for a in terms of b, and hence give a in exact form and determine the size of the blue area.

    Must need to integrate with respect to x and y.
    So one is the integral of
    $x^2$
    And one is the integral of
    $\sqrt x$

    Equate the two and you get the answer - but I don't so I must be wrong, but how am I wrong?
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    It's not the integral of Sqrt(x), but the integral of Sqrt(y).

    Then you integrate one with respect to y and the other as you would normally with respect to x.
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    Sure, from 1 to b, but I can't seem to avoid a = 1!!
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    Integrate both sides and put in a and b leaving:

    \displaystyle \int_1^a \{x^2\} dx = \left[ \frac{x^3}{3} \right]_1^a

    \displaystyle \int_b^1 \{\sqrt{y}\} dy = \left[ \frac{{2y^\frac{3}{2}}}{3}\right]_b^1

    But it's wrong!!
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    (Original post by Gmart)
    Integrate both sides and put in a and b leaving:

    \displaystyle \int_1^a \{x^2\} dx = \left[ \frac{x^3}{3} \right]_1^a

    \displaystyle \int_b^1 \{\sqrt{y}\} dy = \left[ \frac{{2y^\frac{3}{2}}}{3}\right]_b^1

    But it's wrong!!
    Why do you think it's wrong?

    It might be helpful to note that b = a^2.
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    Because:
    $a = 1+\sqrt{3}
    and the above just goes to a = 1
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    (Original post by Gmart)
    Because:
    $a = 1+\sqrt{3}
    and the above just goes to a = 1
    I think that the question is just wrong and no such region exists.
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    (Original post by Gmart)
    Integrate both sides and put in a and b leaving:

    \displaystyle \int_1^a \{x^2\} dx = \left[ \frac{x^3}{3} \right]_1^a

    \displaystyle \int_b^1 \{\sqrt{y}\} dy = \left[ \frac{{2y^\frac{3}{2}}}{3}\right]_b^1

    But it's wrong!!
    Surely you should be integrating the second integral from 1 to b, not b to 1?
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    (Original post by TimGB)
    Surely you should be integrating the second integral from 1 to b, not b to 1?
    Yeah, I agree - but that still gets you a=1.
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    (Original post by Zacken)
    Yeah, I agree - but that still gets you a=1.
    Hmm. That's unfortunate. However, with a bit of geometry you can prove that no such a exists greater than 1.
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    (Original post by Zacken)
    I think that the question is just wrong and no such region exists.
    This diagram should help explain why the question is impossible.
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    (Original post by TimGB)
    This diagram should help explain why the question is impossible.
    Oh, that's neat - I wouldn't have thought about that secant line from A to B!
 
 
 
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