Found problems: 2215
2010 Today's Calculation Of Integral, 566
In the coordinate space, consider the cubic with vertices $ O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)$. Find the volume of the solid generated by revolution of the cubic around the diagonal $ OF$ as the axis of rotation.
2009 Today's Calculation Of Integral, 412
Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$.
(1) Find $ I_0,\ I_{\minus{}1},\ I_2$.
(2) Find $ I_1$.
(3) Express $ I_{n\plus{}2}$ in terms of $ I_n$.
(4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$.
(5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results.
You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.
2019 Jozsef Wildt International Math Competition, W. 56
Let $f$, $g$, $h : [a, b] \to \mathbb{R}$, three integrable functions such that:$$\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1$$Then$$\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1$$
2013 Today's Calculation Of Integral, 863
For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$
(1) Find $\lim_{t\rightarrow 0} F(t).$
(2) Find the range of $t$ such that $F(t)\geq 1.$
2022 CMIMC Integration Bee, 12
\[\int_{\pi/4}^{\pi/2} \tan^{-1}\left(\tan^2(x)\right)\sin(2x)\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
2012 Today's Calculation Of Integral, 777
Given two points $P,\ Q$ on the parabola $C: y=x^2-x-2$ in the $xy$ plane.
Note that the $x$ coodinate of $P$ is less than that of $Q$.
(a) If the origin $O$ is the midpoint of the lines egment $PQ$, then find the equation of the line $PQ$.
(b) If the origin $O$ divides internally the line segment $PQ$ by 2:1, then find the equation of $PQ$.
(c) If the origin $O$ divides internally the line segment $PQ$ by 2:1, find the area of the figure bounded by the parabola $C$ and the line $PQ$.
1995 Miklós Schweitzer, 2
Given $f,g\in L^1[0,1]$ and $\int_0^1 f = \int_0^1 g=1$, prove that there exists an interval I st $\int_I f = \int_I g=\frac12$.
1962 Miklós Schweitzer, 7
Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\]
(where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]
2001 District Olympiad, 3
Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have
\[\int_0^1f(P(x))dx=0\]
Prove that $f(x)=0,\ (\forall)x\in [0,1]$.
[i]Mihai Piticari[/i]
2010 Today's Calculation Of Integral, 532
For a curve $ C: y \equal{} x\sqrt {9 \minus{} x^2}\ (x\geq 0)$,
(1) Find the maximum value of the function.
(2) Find the area of the figure bounded by the curve $ C$ and the $ x$-axis.
(3) Find the volume of the solid by revolution of the figure in (2) around the $ y$-axis.
Please find the volume without using cylindrical shells for my students.
Last Edited.
2007 Today's Calculation Of Integral, 196
Calculate
\[\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \]
2014 NIMO Problems, 6
Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?
[i]Proposed by Lewis Chen[/i]
2011 Today's Calculation Of Integral, 690
Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.
2007 Today's Calculation Of Integral, 182
Find the area of the domain of the system of inequality
\[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]
2005 Today's Calculation Of Integral, 79
Find the area of the domain expressed by the following system inequalities.
\[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]
2004 VJIMC, Problem 1
Suppose that $f:[0,1]\to\mathbb R$ is a continuously differentiable function such that $f(0)=f(1)=0$ and $f(a)=\sqrt3$ for some $a\in(0,1)$. Prove that there exist two tangents to the graph of $f$ that form an equilateral triangle with an appropriate segment of the $x$-axis.
2019 LIMIT Category B, Problem 6
Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$, where $a_0,a_1,a_2,a_3$ are constant. Then
$\textbf{(A)}~f(x)\text{ is differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(B)}~f(x)\text{ is not differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$
$\textbf{(C)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0$
$\textbf{(D)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0,a_3=0$
2002 CentroAmerican, 6
A path from $ (0,0)$ to $ (n,n)$ on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its $ 2n\plus{}1$ vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices $ (0,0)$, $ (n,0)$, $ (n,n)$, $ (0,n)$ into two parts with equal area.
2010 Today's Calculation Of Integral, 612
For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality.
\[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]
2000 Harvard-MIT Mathematics Tournament, 15
$$\lim_{n \to \infty} nr\sqrt[2]{1-\cos \frac{2\pi}{n}}=?$$
PEN G Problems, 8
Show that $e=\sum^{\infty}_{n=0} \frac{1}{n!}$ is irrational.
2011 Today's Calculation Of Integral, 674
Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$
[i]2011 Doshisya University entrance exam/Science and Technology[/i]
2009 German National Olympiad, 6
Let a sequences: $ x_0\in [0;1],x_{n\plus{}1}\equal{}\frac56\minus{}\frac43 \Big|x_n\minus{}\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$
2010 Today's Calculation Of Integral, 528
Consider a function $ f(x)\equal{}xe^{\minus{}x^3}$ defined on any real numbers.
(1) Examine the variation and convexity of $ f(x)$ to draw the garph of $ f(x)$.
(2) For a positive number $ C$, let $ D_1$ be the region bounded by $ y\equal{}f(x)$, the $ x$-axis and $ x\equal{}C$. Denote $ V_1(C)$ the volume obtained by rotation of $ D_1$ about the $ x$-axis. Find $ \lim_{C\rightarrow \infty} V_1(C)$.
(3) Let $ M$ be the maximum value of $ y\equal{}f(x)$ for $ x\geq 0$. Denote $ D_2$ the region bounded by $ y\equal{}f(x)$, the $ y$-axis and $ y\equal{}M$.
Find the volume $ V_2$ obtained by rotation of $ D_2$ about the $ y$-axis.
2005 Today's Calculation Of Integral, 55
Evaluate
\[\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)\]