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For now, assume that [math]a > 0[/math].The loop occurs for [math]x \in [a, 5a][/math]. A sample plot when [math]a = 1[/math] is given below:Then, the area in the loop is given by the integral[math]\begin{align*} A &= \displaystyle \int_a^{5a} \Big[\sqrt{\frac{1}{a} (x - a)(x - 5a)^2} - \Big(-\sqrt{\frac{1}{a} (x - a)(x - 5a)^2}\Big)\Big] [/math][math][/math][math]\, dx\\ &= \displaystyle \frac{2}{\sqrt{a}} \int_a^{5a} (5a - x) \sqrt{x - a} [/math][math][/math][math]\, dx. \end{align*} \tag*{}[/math]Now, we use the substitution [math]t = \sqrt{x - a}[/math]. Then [math]x = a [/math][math][/math][math]+ t^2[/math], and we obtain[math]\begin{align*} A &= \displaystyle \frac{2}{\sqrt{a}} \int_0^{2\sqrt{a}} (4a - t^2) \cdot t \cdot (2t [/math][math][/math][math]\, dt)\\ &= \displaystyle \frac{4}{\sqrt{a}} \int_0^{2\sqrt{a}} (4a t^2 - t^4) [/math][math][/math][math]\, dt\\ &= \displaystyle \frac{4}{\sqrt{a}} \Big(\frac{4a}{3} t^3 - \frac{1}{5}t^5\Big) \Bigg|_0^{2\sqrt{a}}\\ &= \displaystyle \frac{256a^2}{15}. \end{align*} \tag*{}[/math]Remark: We get the same answer when [math]a<0[/math], but we have to use [math]|a|[/math] instead of [math]a[/math] in the calculations above.

First of all let me clarify that the next term can be a lot of terms, because theoretically there exists more than one function which can connect these points. Graphically speaking, these 5 points can be joined by infinitely mane curves, each of which corresponds to a different function.I am assuming a polynomial function for finding that particular function. Since I have only 5 points, I can assume that the polynomial has degree four( since the number of unknown coefficients in that case is 5, which is precisely why we can solve the simultaneous equation from the given data).So, let, [math]f(x)=ax^4+bx^3+cx^2+dx+e[/math]Therefore, [math]f(1)=3=a+b+c+d+e[/math][math]f(2)=10=16a+8b+4c+2d+e[/math][math]f(3)=29=81a+27b+9c+3d+e[/math][math]f(4)=66=256a+64b+16c+4d+e[/math][math]f(5)=127=625a+125b+25c+5d+e[/math]Now, after doing a lot of mathematics, which will do nothing but lengthen my answer, we find that, [math]a=c=d=0, b=1 [/math][math][/math][math][/math] and [math]e=2[/math]Thus [math]f(x)=x^3+2[/math]Hence, [math]f(6)=6^3+2=218[/math]Thus 218 is one of the infinitely other numbers in the next position of the series.

Studying at CCRMA has been the most academically fulfilling experience of my life. Like I tell many people who ask me what CCRMA is all about, it's a very interesting cross between Physics, CS, EE, Arts and Design: all in the context of Audio. While some classes do require a tiny bit experience in theory of music, most classes delve into the Audio part from a signal processing perspective applied in various methods (programming, matlab-ing, breadboard-ing etc.,).There are several aspects that set CCRMA apart from most other departments of the university.It's professor-to-student ratio is roughly 1:4. I'm talking about the active MSTs + PhDs + co-terms. If we include the students from the other departments who take classes in any given quarter, the ratio jumps up to about 1:8. Therefore there is a tremendous amount of interaction between the students, the faculty and the admin staff.Ge Wang and Julius O. Smith III are two very good examples of CCRMA's interdisciplinary collaboration. While CCRMA is where both the professors belong, they draw huge crowds from their respective courtesy departments for courses like Music 256a/b and Music 320, 420a/b, 421 a/b.CCRMA has great musical facilities. A recording studio, a 3D studio listening room (with 22 speakers), a stage (16 high fidelity speakers), 4 personal studios, a physical interaction design lab. The department also hosts quarterly concerts and an annual concert called Transitions, all showcasing the talent of CCRMA throughout the year.Most courses have projects as a final exam.The department building is also the previous residence of the President of Stanford! In fact, the previous ballroom is now the class room.CCRMA houses (or housed) super heroes in the fields Digital Audio and Signal Processing. Max Mathews, John Chowning, Julius Smith, Chris Chafe, Jonathan Abel, Marina Bosi, Poppy Crum, Ge Wang to name a few... (I once realized that there were 4 people sitting right around me who had Wiki articles on them). This translates to a number of publications in any given audio-related conference directly from the folks at CCRMA.The culture in the department is also quite laid back. There are some professors who'd take students very frequently to dinner. There are others who spend late nights in the department researching and having long, engaging conversations with students. How about professors who also jam rock n' roll?My time at CCRMA was also supplemented with courses from other departments at Stanford and it was certainly the greatest academic experince of my life. There was literally nothing more I could ask for.Thanks for the A2A!