# Tag Info

## Hot answers tagged calculus

11 votes
Accepted

• 87.4k
4 votes

• 12.1k
3 votes

### Deal with discontinuity in double integrals

$f$ is obviously not well-defined at $(0,0)$, and we can show that $$\lim_{(x,y) \to (0,0)} f(x,y)$$ does not exist for any path in $D$, since if $y = kx$ for some $k \in [0,1]$, we have for $x > 0$...
• 136k
2 votes

• 99.6k
2 votes

### "Sandwich" Quadratic majorizer of even convex function

A counterexample: $$f(x) = \max(|x| - 2, 0) \, ,\\ p(x) = (x-1)^2 \, .$$ $f$ is convex, even, with $f(0) = 0$ and $f(x) \le p(x)$ for all $x \in \Bbb R$. If $q$ is an even quadratic polynomial ...
• 113k
2 votes
Accepted

### A formula for $\pi$

I can think of a few things a mathematician might care about including: beauty, computational complexity, convergence, and sign of the error. Beauty: This is highly subjective, though it oftentimes ...
• 6,443
2 votes

The derivative is $$f'(x)=\frac{\pi x \cos (\pi x)-2 \sin (\pi x)}{x^3}$$ Because of the leading $x$, the solution will be closer and closer to $\left(n+\frac{1}{2}\right)$. Let $x=\left(n+\frac{1}... • 261k 2 votes ### How to evaluate$\int_0^1 \frac{x^n}{x^2+x+1} \,dx$Note that$x^2+x+1=(x-e^{i\frac{2\pi}3})(x-e^{-i\frac{2\pi}3} )$and $$\int_0^1 \frac{x^n}{x-a}dx =a^n\ln\frac{a-1}a+\sum_{k=0}^{n-1}\frac{a^k}{n-k}$$ Then, with$a= e^{i\frac{2\pi}3}\begin{align} &... • 97.5k 2 votes ### definite integral\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx\$

Note that \begin{align} I=&\int_{0}^{\frac{\pi}{4}} \frac{\sin^2x\cos^2x}{\sin^3x+\cos^3x}dx\\ =&\int_{0}^{\frac{\pi}{4}} \frac{(\sin x\cos x)^2(\sin x+\cos x)}{(\sin x+\cos x)^2(\sin^2x-\sin ...
• 97.5k

Only top scored, non community-wiki answers of a minimum length are eligible