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Math Help - Strange question involving definite integrals

  1. #1
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    Strange question involving definite integrals

    Let .
    Find
    and


    How can I go about solving this? I've done examples on definite integrals and did my note-reading but nowhere does it explain how to deal with this kind of problem. Any help is appreciated!
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    Quote Originally Posted by Archduke01 View Post
    Let .
    Find
    and


    How can I go about solving this? I've done examples on definite integrals and did my note-reading but nowhere does it explain how to deal with this kind of problem. Any help is appreciated!
    properties of definite integrals you should already be familiar with ...

    \int_a^b f(x) \pm g(x) \, dx = \int_a^b f(x) \, dx \pm \int_a^b g(x) \, dx

    \int_a^b k \cdot f(x) \, dx = k \int_a^b f(x) \, dx

    \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx

    \int_a^b f(x) \, dx + \int_b^c f(x) \, dx + \int_c^d f(x) \, dx = \int_a^d f(x) \, dx
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    How do I use the properties to solve the problems?
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    use this property plus the given values for the various definite integrals of f(x) ...

    \int_a^b f(x) \, dx + \int_b^c f(x) \, dx + \int_c^d f(x) \, dx = \int_a^d f(x) \, dx

    ... to find \int_{5.5}^8 f(x) \, dx
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  5. #5
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    Thank you! I got the first one.

    For the second, I notice that the upper limit is actually a smaller value than the lower. So that means I can flip them and make the integral negative? Even so, the 6f(x)-10 throws me off... Do I replace the f(x) with -10 (which was the answer to the previous question) and solve?
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  6. #6
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    Quote Originally Posted by Archduke01 View Post
    Thank you! I got the first one.

    For the second, I notice that the upper limit is actually a smaller value than the lower. So that means I can flip them and make the integral negative? Even so, the 6f(x)-10 throws me off... Do I replace the f(x) with -10 (which was the answer to the previous question) and solve?
    properties, properties, properties ...

    \int_8^{5.5} 6f(x) - 10 \, dx = \int_8^{5.5} 6f(x) \, dx - \int_8^{5.5} 10 \, dx =  6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx <br />
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    6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx

    Thank you, but do I flip the a and b values now and make the integrals negative/positive? Because one of the properties state that.
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    Please respond. I don't know how to proceed.
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    6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx = -6 \int_{5.5}^8 f(x) \, dx + \int_{5.5}^8 10 \, dx

    now finish it.
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  10. #10
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    6 \int_8^{5.5} f(x) \, dx - \int_8^{5.5} 10 \, dx = -6 <br />
\int_{5.5}^8 f(x) \, dx + \int_{5.5}^8 10 \, dx<br />

    Since
    -6 \int_{5.5}^8 f(x) = -10, the equation becomes

    -6 * -10 + \int_{5.5}^8 10 \, dx

    What can I do to the second one to change that into a number too?
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  11. #11
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    Wrong pleace, sorry.
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  12. #12
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    I'm sorry, I don't understand. How was that formula relevant to my question?
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