Sophia Rossi

I'm studying calculus and I'm learning about volumes of solids of revolution. I know there are two methods: **Disk method:** $V = \pi \int_a^b [R(x)]^2 \, dx$ when rotating around the $x$-axis

I'm preparing for IIT-JEE and I need to understand the classic proof that $\sqrt{2}$ is irrational. I've seen the proof but I have some doubts about the logic. The proof assumes $\sqrt{2} = p/q$ in lowest terms, then squares both sides to get $2q^2 = p^2$. Then it argues that $p$ must be even, so $p = 2k$, and substituting gives $q^2 = 2k^2$, so $q$ is also even, contradicting the assumption that $p/q$ is in lowest terms.

I'm studying probability and I understand the formula for Bayes' Theorem: $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

I'm studying multivariable calculus and I've encountered Stokes' theorem: $$\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\ abla \times \mathbf{F}) \cdot \mathbf{n} \, dS$$

I need to integrate $\int \frac{x^2 + 2x + 3}{(x-1)(x+2)^2} \, dx$ and I know I need partial fractions first. The general decomposition is:

I'm studying combinatorics and I'm stuck on a problem involving permutations with restrictions.

Problem: \"How many ways can 5 distinct mathematics books and 3 distinct physics books be arranged on a shelf if all books of the same subject must be together?\"

I'm studying linear algebra and I need a clear method for checking whether a set of vectors is linearly independent. Formally, vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$ are linearly independent if the only solution to: