This problem is from the 7th edition of the book "Calculus and Analytic Geometry" by George Thomas and Ross Finney. It is problem number 3 of section 5.4 Problem: Find the volume generated when the region bounded by the given curves and lines is revolved about the x-axis. (Note: $x = 0$ is the y-...

Problem: Find the volume generated when the region bounded by the given curves and lines is revolved about the x-axis using the disk method. Then find it using the cylindrical shell method and verify that they produce the same result. \begin{align*} y &= x^2 \\ x &= 1 \\ x &= 2 \\ \end{align*} An...

Let $p_i$ be the i th prime. Define $$ \zeta_H(s) = \prod_{i = 1}^{\infty} \frac{p_{2i}^s}{p_{2i}^s - 1} $$ Consider it's analytic continuation. ( feel free to explain that analytic continuation ) Where are the zero's and poles of this function ? Does this function have all it's nontrivial zero's...

Each radical $\sqrt[k+1]a$ forms an array of the variables $(p, p^2,\dots, p^k),$ wherein there is not a problem to exclude them via a simple linear system. In particular, for the expression $x=\sqrt a + \sqrt[3]b$ easily to get $$ \begin{cases} p+q-x = 0\\ p^2=a\\ q^3 = b \end{cases}\Rightarrow ...

Consider the sequence $f(i)$ for integer $i>0$ : $f(1) = f(2) = 1 $ $f(3) = 2$ define the remaining by $f(2n) = f(2n-1) + \frac{f(2n-2)}{f(n)}$ $f(2n+1) = f(2n) + \frac{2n-1}{f(n)}$ How fast does this function grow ? What are the best asymptotics ? How to answer such similar questions in general ...