Found problems: 1687
2002 Flanders Math Olympiad, 3
show that $\frac1{15} < \frac12\cdot\frac34\cdots\frac{99}{100} < \frac1{10}$
2009 Today's Calculation Of Integral, 512
Evaluate $ \int_0^{n\pi} \sqrt{1\minus{}\sin t}\ dt\ (n\equal{}1,\ 2,\ \cdots).$
2011 Today's Calculation Of Integral, 749
Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.
2023 CMIMC Integration Bee, 5
\[\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2012 Today's Calculation Of Integral, 841
Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$
1980 USAMO, 3
Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.
2012 Today's Calculation Of Integral, 770
Find the value of $a$ such that :
\[101a=6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]
2005 Romania Team Selection Test, 3
Let $P$ be a polygon (not necessarily convex) with $n$ vertices, such that all its sides and diagonals are less or equal with 1 in length. Prove that the area of the polygon is less than $\dfrac {\sqrt 3} 2$.
2009 ISI B.Stat Entrance Exam, 2
Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that
\[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]
2011 Today's Calculation Of Integral, 682
On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$.
If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then
evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$
[i]2011 Waseda University of Education entrance exam/Science[/i]
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]
1983 Miklós Schweitzer, 12
Let $ X_1,X_2,\ldots, X_n$ be independent, identically distributed, nonnegative random variables with a common continuous distribution function $ F$. Suppose in addition that the inverse of $ F$, the quantile function $ Q$, is also continuous and $ Q(0)=0$. Let $ 0=X_{0: n} \leq X_{1: n} \leq \ldots \leq X_{n: n}$ be the ordered sample from the above random variables. Prove that if $ EX_1$ is finite, then the random variable \[ \Delta = \sup_{0\leq y \leq 1} \left| \frac 1n \sum_{i=1}^{\lfloor ny \rfloor +1} (n+1-i)(X_{i: n}-X_{i-1: n})- \int_0^y (1-u)dQ(u) \right|\] tends to zero with probability one as $ n \rightarrow \infty$.
[i]S. Csorgp, L. Horvath[/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
\[\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \]
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.
\[x\geq 0,\ y\geq 0,\ x^{\frac{1}{p}}+y^{\frac{1}{p}} \leq 1\ (p=1,2,\cdots)\]