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Thread: Problems with some integration

  1. #16
    Grand Panjandrum
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    Quote Originally Posted by scorpion007
    Here's also another question i couldnt solve. Its multiple choice. See if you can get it.

    $\displaystyle f$ is a continuous, differentiable function with $\displaystyle f(x) = f(-x)$ such that $\displaystyle f(x) \ge 0$ for $\displaystyle x \in [0, a]$ and $\displaystyle f(x) < 0$ for $\displaystyle x \in (a, b]$. It is also known that $\displaystyle \int^{b}_{0} f(x) dx = 0$ and $\displaystyle \int^{a}_{0} f(x) dx = A$. Which of the following is false?

    C: $\displaystyle -\int^{b}_{a} f(x) dx + \int^{a}_{0} f(x) dx - \int^{-a}_{0} f(x) dx = 3A$(
    Is true:

    $\displaystyle
    -\int^{b}_{a} f(x) dx + \int^{a}_{0} f(x) dx - \int^{-a}_{0} f(x) dx$$\displaystyle = -\int^{b}_{a} f(x) dx + \int^{a}_{0} f(x) dx + A
    $

    by part B.

    $\displaystyle
    = -\int^{b}_{a} f(x) dx + A + A
    $
    Middle integral is equal to $\displaystyle A$ is given in the question

    Finally:

    $\displaystyle
    -\int^{b}_{a} f(x) dx=\int^{a}_{b} f(x) dx=$$\displaystyle \int^{a}_{0} f(x) dx-\int^{b}_{0} f(x) dx=A-0=A
    $.

    So:

    $\displaystyle
    -\int^{b}_{a} f(x) dx + \int^{a}_{0} f(x) dx - \int^{-a}_{0} f(x) dx=3A
    $.

    RonL
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  2. #17
    Grand Panjandrum
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    Quote Originally Posted by scorpion007
    Here's also another question i couldnt solve. Its multiple choice. See if you can get it.

    $\displaystyle f$ is a continuous, differentiable function with $\displaystyle f(x) = f(-x)$ such that $\displaystyle f(x) \ge 0$ for $\displaystyle x \in [0, a]$ and $\displaystyle f(x) < 0$ for $\displaystyle x \in (a, b]$. It is also known that $\displaystyle \int^{b}_{0} f(x) dx = 0$ and $\displaystyle \int^{a}_{0} f(x) dx = A$. Which of the following is false?

    D: $\displaystyle \Bigl\vert \int^{b}_{a} f(x) dx \Bigr\vert + \Bigl\vert \int^{a}_{0} f(x) dx \Bigr\vert = 2A$
    Is true, as:

    $\displaystyle
    \int^{b}_{a} f(x) dx=\int^{b}_{0} f(x) dx-\int^{a}_{0} f(x) dx =0-A=-A
    $

    as the two integrals on the RHS are given,

    and:

    $\displaystyle \int^{a}_{0} f(x) dx=A$

    is given.

    But as $\displaystyle f(x) \ge 0$ on $\displaystyle [0,a]$, $\displaystyle A \ge 0$ so:

    $\displaystyle \Bigl\vert \int^{b}_{a} f(x) dx \Bigr\vert + \Bigl\vert \int^{a}_{0} f(x) dx \Bigr\vert =|-A|+|A|= 2A$

    RonL
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  3. #18
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    Quote Originally Posted by scorpion007
    Here's also another question i couldnt solve. Its multiple choice. See if you can get it.

    $\displaystyle f$ is a continuous, differentiable function with $\displaystyle f(x) = f(-x)$ such that $\displaystyle f(x) \ge 0$ for $\displaystyle x \in [0, a]$ and $\displaystyle f(x) < 0$ for $\displaystyle x \in (a, b]$. It is also known that $\displaystyle \int^{b}_{0} f(x) dx = 0$ and $\displaystyle \int^{a}_{0} f(x) dx = A$. Which of the following is false?

    E: $\displaystyle \int^{-a}_{-b} f(x) dx = 0$
    Is obviously false.

    RonL
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  4. #19
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    thanks a lot!

    I have another question:

    Given that there is 12.5 kg (12500 g) or chlorine at time t=0 in a pool, use the Euler's method of approximation with h=30 to solve this differential equation:

    Q is amount of chlorine in grams
    t is in minutes.
    $\displaystyle \frac{dQ}{dt} = \frac{-5Q}{5000-4t} $

    Basically I have to fill a table like so:
    Code:
    | n | $\displaystyle t_n$ | $\displaystyle Q_n$  |
    ------------------
    | 0 | 0  | 12500 |
    | 1 |    |       |
    ...
    | 5 |    |       |
    Any suggestions?
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  5. #20
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    Quote Originally Posted by CaptainBlack
    Looks like making the substitution $\displaystyle \cos(u)=x$, and then
    integrating by parts.

    RonL
    I think the question is asking for:

    $\displaystyle
    \int_0^a \arccos(x)\ dx
    $.

    Let $\displaystyle \cos(u)=x$, then $\displaystyle -\sin(u)\ \frac{du}{dx}=1$ so the integral becomes:

    $\displaystyle
    -\int_{\pi/2}^{\arccos(a)} u\ \sin(u)\ du
    $.

    Then proceed with integration by parts.

    RonL
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  6. #21
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    Quote Originally Posted by scorpion007
    thanks a lot!

    I have another question:

    Given that there is 12.5 kg (12500 g) or chlorine at time t=0 in a pool, use the Euler's method of approximation with h=30 to solve this differential equation:

    Q is amount of chlorine in grams
    t is in minutes.
    $\displaystyle \frac{dQ}{dt} = \frac{-5Q}{5000-4t} $

    Basically I have to fill a table like so:
    Code:
    | n | $\displaystyle t_n$ | $\displaystyle Q_n$  |
    ------------------
    | 0 | 0  | 12500 |
    | 1 |    |       |
    ...
    | 5 |    |       |
    Any suggestions?
    You need a table like:

    $\displaystyle
    \begin{array}{c|ccc}
    n&t_n&Q'_n&Q_n\\ \hline
    0&0&-12.5&12500\\
    1&30&-12.81&12125\\
    :&:&:&:
    \end{array}
    $

    where $\displaystyle Q_n=Q_{n-1}+h\ Q'_{n-1}$, $\displaystyle Q'_n=\frac{-5Q_n}{5000-4t_n}$, and $\displaystyle t_n=n.h=n.30$

    RonL
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  7. #22
    Senior Member
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    thanks very much
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