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Thread: area of the region

  1. #1
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    area of the region

    Decide whether to integrate with respect to x or y. Then find the area of the region

    $\displaystyle y = 7 \cos x, y = (6 sec(x))^2, x = -\pi / 4, x = \pi / 4 $

    i used the formula: $\displaystyle A = \int_a^b(f(x)-g(x)) dx $

    $\displaystyle A = \int_a^b ((6 sec(x))^2 - 7 cosx)dx$

    $\displaystyle a = -\pi / 4 , b = -\pi / 4 $

    not sure what to do next
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  2. #2
    fgn
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    Quote Originally Posted by viet View Post
    $\displaystyle A = \int_a^b ((6 sec(x))^2 - 7 cosx)dx$

    $\displaystyle a = -\pi / 4 , b = -\pi / 4 $
    You almost had it.
    But $\displaystyle \int_a^bf(x)\,dx = 0 $ if $\displaystyle a = b $ by the fundamental theorem of calculus.
    If you change it to this it should work.
    $\displaystyle A = \int_{-\frac{\Pi}{4}}^{\frac{\Pi}{4}}((6 sec\,(x))^2 - 7 cos\,(x))\,dx$
    You could also exploit the symmetry to get
    $\displaystyle A = 2\int_0^{\frac{\Pi}{4}}((6 sec\,(x))^2 - 7 cos\,(x))\,dx $
    Last edited by fgn; Dec 7th 2006 at 02:05 PM. Reason: typo
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  3. #3
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    Quote Originally Posted by fgn View Post
    If you change it to this it should work.
    $\displaystyle A = \int_{-\frac{\Pi}{4}}^{\frac{\Pi}{4}}((6 sec\,(x))^2 - 7 cos\,(x))\,dx$
    i understand that part, but not sure how to get the answer from here
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  4. #4
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    Viet isn't the only one looking for help on this particular problem. I know a couple of others are as well. Can anyone help us through this problem step by step?
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  5. #5
    fgn
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    Quote Originally Posted by viet View Post
    i understand that part, but not sure how to get the answer from here
    So far so good!

    It's good if you try to learn the derivatives of the elementary functions.
    $\displaystyle \frac{d}{dx}(tan\,(x)) = \frac{sin^{\,2}(x) + cos^{\,2}(x)}{cos^{\,2}(x)} = sec^{\,2}(x) = tan^{\,2}(x) + 1 $

    From this you get $\displaystyle \int sec^{\,2}(x)\,dx = tan\,(x) + C $

    And your integral becomes
    $\displaystyle \int((6 sec\,(x))^2 - 7 cos\,(x))\,dx = 36tan\,(x) - 7sin\,(x) + C$

    I hope you can take it from here.
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