Problems with some integration

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July 22nd 2006, 09:04 PM
CaptainBlack

Quote:

Originally Posted by scorpion007

Here's also another question i couldnt solve. Its multiple choice. See if you can get it.

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

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

Is true:

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

by part B.

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

Finally:

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

So:

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

RonL
July 22nd 2006, 09:21 PM
CaptainBlack

Quote:

Originally Posted by scorpion007

Here's also another question i couldnt solve. Its multiple choice. See if you can get it.

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

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

Is true, as:

$ \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:

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

is given.

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

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

RonL
July 22nd 2006, 09:23 PM
CaptainBlack

Quote:

Originally Posted by scorpion007

Here's also another question i couldnt solve. Its multiple choice. See if you can get it.

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

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

Is obviously false.

RonL
July 23rd 2006, 10:07 PM
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.
$\frac{dQ}{dt} = \frac{-5Q}{5000-4t}$

Basically I have to fill a table like so:

Code:

| n | | | ------------------ | 0 | 0 | 12500 | | 1 | | | ... | 5 | | |

Any suggestions?
July 24th 2006, 04:29 AM
CaptainBlack

Quote:

Originally Posted by CaptainBlack

Looks like making the substitution $\cos(u)=x$ , and then
integrating by parts.

RonL

I think the question is asking for:

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

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

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

Then proceed with integration by parts.

RonL
July 24th 2006, 04:47 AM
CaptainBlack

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.
$\frac{dQ}{dt} = \frac{-5Q}{5000-4t}$

Basically I have to fill a table like so:

Code:

| n | | | ------------------ | 0 | 0 | 12500 | | 1 | | | ... | 5 | | |

Any suggestions?

You need a table like:

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

RonL
July 25th 2006, 03:16 AM
scorpion007

thanks very much

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