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  1. #1
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    integrals

    Let .

    Find ________

    and _________
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  2. #2
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    Quote Originally Posted by tbenne3 View Post
    Let .

    Find ________

    and _________

    Remember that

    $\displaystyle \int_a^b{f(x)\,dx} = \int_a^c{f(x)\,dx} + \int_c^b{f(x)\,dx}$ for $\displaystyle a \leq c \leq b$.

    So $\displaystyle \int_{-4}^{0.5}{f(x)\,dx} = \int_{-4}^{-2.5}{f(x)\,dx} + \int_{-2.5}^{-1}{f(x)\,dx} + \int_{-1}^{0.5}{f(x)\,dx}$.
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  3. #3
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    Hello, tbenne3!

    Did you make a sketch?


    Let: .$\displaystyle \int^{0.5}_{-4}f(x)\,dx \:=\:4 \qquad \int^{-2.5}_{-4}f(x)\,dx\;=\;4 \qquad \int^{0.5}_{-1}f(x)\,dx \;=\;1 $
    These are areas "under" the curve.

    With a little thought, we see that the graph looks like this:
    Code:
                                          |
               .*.                        |
            .*:::::*.                     |
           *::: 4 :::*               ..* *|.
           :::::::::::           -1.*:: 1 |:*.
      - - * - - - - - * - - - - - * - - - + - * - - - - -
         -4        -2.5 *: -1 ::*         | -0.5
                           * *            |
                                          |

    $\displaystyle \text{Find: }\;\int^{-1}_{-2.5}f(x)\,dx$

    $\displaystyle \text{Find: }\;\int^{-2.5}_{-1}\bigg[4f(x) - 4\bigg]\,dx $

    Can you work out these areas now?

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  4. #4
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    Quote Originally Posted by Prove It View Post
    Remember that

    $\displaystyle \int_a^b{f(x)\,dx} = \int_a^c{f(x)\,dx} + \int_c^b{f(x)\,dx}$ for $\displaystyle a \leq c \leq b$.

    So $\displaystyle \int_{-4}^{0.5}{f(x)\,dx} = \int_{-4}^{-2.5}{f(x)\,dx} + \int_{-2.5}^{-1}{f(x)\,dx} + \int_{-1}^{0.5}{f(x)\,dx}$.
    so where do I go from there?
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  5. #5
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    Quote Originally Posted by Soroban View Post
    Hello, tbenne3!

    Did you make a sketch?


    These are areas "under" the curve.

    With a little thought, we see that the graph looks like this:
    Code:
                                          |
               .*.                        |
            .*:::::*.                     |
           *::: 4 :::*               ..* *|.
           :::::::::::           -1.*:: 1 |:*.
      - - * - - - - - * - - - - - * - - - + - * - - - - -
         -4        -2.5 *: -1 ::*         | -0.5
                           * *            |
                                          |

    Can you work out these areas now?

    I don't understand what it's asking for
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  6. #6
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    Quote Originally Posted by tbenne3 View Post
    so where do I go from there?
    You should realise that you've been given 3 of those 4 integrals. So substitute them and solve for the 4th.
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  7. #7
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    Quote Originally Posted by Prove It View Post
    You should realise that you've been given 3 of those 4 integrals. So substitute them and solve for the 4th.
    I got the first but must be doing something wrong for the second
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    Well that's not surprising considering that I haven't taught you how to do Part 2 yet...

    $\displaystyle \int_{-1}^{-2.5}{4f(x) - 4\,dx} = \int_{-1}^{-2.5}{4f(x)\,dx} - \int_{-1}^{-2.5}{4\,dx}$

    $\displaystyle = 4\int_{-1}^{-2.5}{f(x)\,dx} - \int_{-1}^{-2.5}{4\,dx}$

    $\displaystyle = -4\int_{-2.5}^{-1}{f(x)\,dx} + \int_{-2.5}^{-1}{4\,dx}$.
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  9. #9
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    Quote Originally Posted by Prove It View Post
    Well that's not surprising considering that I haven't taught you how to do Part 2 yet...

    $\displaystyle \int_{-1}^{-2.5}{4f(x) - 4\,dx} = \int_{-1}^{-2.5}{4f(x)\,dx} - \int_{-1}^{-2.5}{4\,dx}$

    $\displaystyle = 4\int_{-1}^{-2.5}{f(x)\,dx} - \int_{-1}^{-2.5}{4\,dx}$

    $\displaystyle = -4\int_{-2.5}^{-1}{f(x)\,dx} + \int_{-2.5}^{-1}{4\,dx}$.
    can't figure out why i'm not getting it right.. what are you plugging in for f(x)?
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