Daedalus

Given the equation for a circle offset from the origin such that the bottom of the circle touches the [math]x[/math] axis we get: [math]x^2+(y-r)^2=r^2[/math] Solving for [math]y[/math], we get: [math](y-r)^2=r^2-x^2[/math] [math]y-r=\sqrt{r^2-x^2}[/math] [math]y=\sqrt{r^2-x^2}+r[/math] Because square roots are involved, there are two functions to consider. The top half of the circle: [math]y=r+\sqrt{r^2-x^2}[/math] or the bottom half: [math]y=r-\sqrt{r^2-x^2}[/math] Since your diagram shows the bottom half, we'll go with [math]f(x)=r-\sqrt{r^2-x^2}[/math] Now, we simply integrate the function on a specified interval [[math]a,b[/math]] [math]\int_{a}^{b}r-\sqrt{r^2-x^2}\, dx[/math] or, we can solve for the antiderivative to get: [math]\int r-\sqrt{r^2-x^2}\, dx = r x - \frac{1}{2}\, x\, \sqrt{r^2 - x^2} + \frac{1}{2}\, r^2\, \text{tan}^{-1}\left (\frac{x \sqrt{r^2 - x^2}}{x^2-r^2}\right )+C[/math]

Although you worded incorrectly, I believe I know what you're asking. However, there isn't an equation that is defined only for a quarter circle without restricting it to some interval such as [latex]F\left(\theta\right)=\left \langle r\,\text{cos}\,\theta,\ r\,\text{sin}\,\theta \right \rangle \ \ \left \{r,\theta\in\mathbb{R}\,\vert \, 0 \le \theta \le 45^\circ \right \}[/latex] Here we have the parametric equation for a circle which makes defining the [latex]x[/latex] and [latex]y[/latex] coordinates easier as you are wanting to restrict the equation according to a specific angle. The interval [[latex]0,45^\circ[/latex]] is an arbitrary choice that is dependent upon our parametrization of the circle or vice versa. We can achieve the same restriction using an interval [[latex]0,1[/latex]]. [latex]F\left(t\right)=\left \langle r\,\text{cos}\left(45 t\right),\ r\,\text{sin}\left(45 t\right) \right \rangle \ \ \left \{r,t\in\mathbb{R}\,\vert \, 0 \le t \le 1 \right \}[/latex] Given the equation for a circle that Imatfaal has provided [math]x^2+y^2=r^2[/math], we can demonstrate that our parametric equation does indeed satisfy the conditions of the equation for a circle centered at the origin - (0,0). [math]x=r\,\text{cos}\,\theta[/math] [math]y=r\,\text{sin}\,\theta[/math] [math]\left(r\,\text{cos}\,\theta\right)^2+\left(r\,\text{sin}\,\theta\right)^2=r^2[/math] [math]r^2\,\text{cos}^2\theta+r^2\,\text{sin}^2\theta=r^2[/math] [math]r^2\left(\text{cos}^2\theta+\text{sin}^2\theta\right)=r^2[/math] Use the Pythagorean trigonometric identity to simplify the equation. [math]\text{cos}^2\theta+\text{sin}^2\theta=1[/math] [math]r^2\left(1\right)=r^2[/math] [math]r=r[/math]

ROFL!!!

It can always get worse....

Here is a poem I wrote for my ex-wife when we first seperated. Now, I am happily divorced, but the poem reflects my feelings for her at the time. Unlocked Cage Throughout the years I have been cruel I locked you up just like a fool Within my heart I caged you in A love so true should never end It is hard for me to let you go It feels like losing part of my soul I know you just need time to see That you can share a life with me I should have seen your colors fade I should have heard your love song wane I should have been a better friend To be with you until the end But now I must set you free To release you from your misery I unlocked the cage I put you in So spread your wings and fly again

Lies upon lies!!!

Now this is how you scare the crap out of someone!!!

When a particle decays, such as the neutron, and you get back other particles such as a proton, electron, and an electron antineutrino, it's easy to come to the conclusion that such particle is similar to the hydrogen atom because of the particles that you get back are also found in such atom. However, you have to keep in mind that energy is transformed. So, a particle such as a neutron can decay releasing its energy, which is transformed into other particles that can be found in atoms such as hydrogen. This does not mean that a neutron is a hydrogen atom smashed to the point that it becomes a neutron, only that the energy released from such decay is transformed into particles that can also be found in the hydrogen atom. As for point particles, I'm by no means an expert on particle physics, but from what I understand that is how the mathematics treats the particles. Perhaps, someone more knowledgeable in that field can weigh in on the topic.

Are you having a bad day at work? Well, then read this: It could be worse...

I don't quite get the joke Moontanman... If that happened to me, I'm pretty sure humor would be the last thing on my mind.

Seems fitting given the above posts:

Here is a sneak preview of the next song I'm writing: https://drive.google.com/file/d/0BwscqcqDS8fVMnJTMU1NRHA3MVE/edit?usp=sharing Since the song isn't finished, I've uploaded the audio file to Google Docs. So, if you want to listen to it, you will need to click on the above link and download the audio file. Again, I will leave the song on Google Docs for a few weeks or until I have it finished and published to youtube. Enjoy!!!

Thank you Lightmeow! Although I wouldn't mind being a mainstream artist (the pay is better), all of my music comes from the heart and each song has meaning.

I haven't been able to spend much time with my boys, and I've been missing them a lot here lately. So, I wrote this song for them. I wanted to embellish the melody on the second change up, but the song contains 7 independent voices. This creates a huge drain on my PC's resources and Pro Tools would not play the faster parts correctly (not enough processing power to do so). However, the song did turn out beautifully. So, I'll see what I can do in the studio.

I agree completely about the lack of dynamics. I'm am relatively new to Pro Tools. So, I am still learning all the controls. However, I am working with a professional recording studio, Evermind, in producing quality recordings of my music - dynamics and all. I'll keep you all posted on the progress.

Thank you Arc. Hopefully, I will have my next song done this week. It's coming along nicely.

In addition to thanking Imatfaal for putting together such a wonderful tutorial that will hopefully be put to use, I would also like to add that you can nest quotes. The forum software used to do this automatically when you quoted someone who also quoted someone else. To nest quotes, you simply copy and paste the quoted text inside the quote. Like in an essay - make it clear you are paraphrasing. It is rather external to this thread as a section of prose that is paraphrasing another's ideas never ever goes in quotation marks. As Stringjunky says below it might be seen as deceptive if a paraphrase is made to appear as if it were a direct quote and it is good posting practice to overtly mention that you are re-wording another persons text (see what I did there - metaparaphrasing) The behaviour of the cursor and the software doing "its own thing" is one of the reasons we decided a thread was necessary. I tried to list things that (nearly) always work - but if there are problems with any of the methods let me know and I will attempt a workaround. Of course, this involves you pressing the quote button for both posts, copying the quoted text, and then pasting one quote inside the other.

Happy belated Pi Day!!!!

Anyone who has been married or in a relationship for a very long time will appreciate this one: A man and woman had been married for more than 60 years. They had shared everything. They had talked about everything. They had kept no secrets from each other except that the little old woman had a shoe box in the top of her closet that she had cautioned her husband never to open or ask her about. For all of these years, he had never thought about the box, but one day the little old woman got very sick and the doctor said she would not recover. In trying to sort out their affairs, the little old man took down the shoe box and took it to his wife's bedside. She agreed that it was time that he should know what was in the box. When he opened it, he found two crocheted dolls and a stack of money totalling $95,000. He asked her about the contents. 'When we were to be married,' she said, ' my grandmother told me the secret of a happy marriage was to never argue. She told me that if I ever got angry with you, I should just keep quiet and crochet a doll.' The little old man was so moved; he had to fight back tears. Only two Precious dolls were in the box. She had only been angry with him two Times in all those years of living and loving. He almost burst with Happiness. 'Honey,' he said, 'that explains the dolls, but what about all of this money? Where did it come from?' 'Oh,' she said, 'that's the money I made from selling the dolls.'

That's awesome CaptainPanic. Thanks for posting the Ted talk video.

After reading a post on Facebook from IFLS about the new paper microscope, FoldScope, I must say that I'm impressed! I want one!!!

Well this year is definitely my year for writing music. "G Minor Blues" is my third composition since January and my first song to use blues scales, and I must say it's one of my finest works. It has seven independent and intertwining voices. Enjoy!!!

Here is a sneak preview of the next song I'm writing. The link is to a file stored in my Google Docs. So, you will have to download the .WMA file to hear it. https://drive.google.com/file/d/0BwscqcqDS8fVRC1xRVFHcFplUU0/edit?usp=sharing I'll post it to youtube once I get it finished. This file won't be listed in my docs for long. I'll leave it up for a few weeks. Enjoy this small excerpt!

Sorry StringJunky, I noticed a horrible hiss in the audio due to youtube converting the audio format. I ran a high-pass filter and it did reduce the hiss to a tolerable level. I'm sorry for the inconvenience, but it is driving me insane... The problem is with the "Cathedral Organ Bourd Pedal" sound that I am using in Pro Tools. When you listen to the song, you will hear the wind blowing through the pipes. It is this part of the sound that gets mangled on the high end. Unfortunately, I cannot edit it out in Pro Tools. I did use an EQ, but then youtube defeated the purpose. Luckily, Audacity came to save the day and removed a lot of the hiss... Here is a link to the .WMA file of the original recording: https://drive.google.com/file/d/0BwscqcqDS8fVb1Biek5uRWEyeXc/edit?usp=sharing Well, if you think about it, you can enjoy the song and still provide criticism on parts you might want me to rewrite. And on that note, I'm going to bed...