Olympiad problems

2012 Today's Calculation Of Integral, 849

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

2012 Today's Calculation Of Integral, 800

Tags: calculus , integration , trigonometry , function , calculus computations

For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$

2010 Today's Calculation Of Integral, 592

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

2010 Today's Calculation Of Integral, 668

Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations

Consider two curves $y=\sin x,\ y=\sin 2x$ in $0\leq x\leq 2\pi$. (1) Let $(\alpha ,\ \beta)\ (0<\alpha <\pi)$ be the intersection point of the curves. If $\sin x-\sin 2x$ has a local minimum at $x=x_1$ and a local maximum at $x=x_2$, then find the values of $\cos x_1,\ \cos x_1\cos x_2$. (2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of $\alpha \leq x\leq \pi$ for the figure about the line $y=-1$. [i]2011 Kyorin　University entrance exam/Medicine [/i]

2011 Today's Calculation Of Integral, 747

Tags: calculus , integration , trigonometry , inequalities , calculus computations

Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$

2011 Today's Calculation Of Integral, 714

Tags: calculus , integration , geometry , calculus computations

Find the area enclosed by the graph of $a^2x^4=b^2x^2-y^2\ (a>0,\ b>0).$

2011 Today's Calculation Of Integral, 684

Tags: calculus , integration , geometry , function , parabola , calculus computations , conic

On the $xy$ plane, find the area of the figure bounded by the graphs of $y=x$ and $y=\left|\ \frac34 x^2-3\ \right |-2$. [i]2011 Kyoto University entrance exam/Science, Problem 3[/i]

2010 Today's Calculation Of Integral, 614

Tags: calculus , integration , ratio , trigonometry , calculus computations

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2013 Today's Calculation Of Integral, 898

Tags: calculus , integration , calculus computations , logarithm

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

2007 Today's Calculation Of Integral, 239

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\pi} \sin (\pi \cos x)\ dx.$

2007 Today's Calculation Of Integral, 191

Tags: calculus , integration , geometry , limit , ratio , geometric series , calculus computations

(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$ (2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$

2012 Today's Calculation Of Integral, 788

Tags: calculus , integration , function , analytic geometry , calculus computations , logarithm

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

2013 Today's Calculation Of Integral, 894

Tags: calculus , integration , geometry , calculus computations , logarithm

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

2008 ISI B.Stat Entrance Exam, 6

Tags: limit , integration , calculus , function , real analysis , calculus computations , logarithm

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

2007 Today's Calculation Of Integral, 190

Tags: calculus , integration , calculus computations

In $xyz$ space, let $l$ be the segment joining two points $(1,\ 0,\ 1)$ and $(1,\ 0,\ 2),$ and $A$ be the figure obtained by revolving $l$ around the $z$ axis. Find the volume of the solid obtained by revolving $A$ around the $x$ axis. Note you may not use double integral.

2012 Today's Calculation Of Integral, 799

Tags: calculus , integration , limit , induction , strong induction , calculus computations , logarithm

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

2009 Today's Calculation Of Integral, 454

Tags: calculus , integration , trigonometry , calculus computations

Let $ a$ be positive constant number. Evaluate $ \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.$

2009 Today's Calculation Of Integral, 424

Tags: calculus , integration , geometry , limit , trigonometry , calculus computations

Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

2009 Today's Calculation Of Integral, 470

Tags: calculus , integration , parabola , calculus computations , conic , geometry

Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.

2010 Today's Calculation Of Integral, 608

Tags: calculus , integration , calculus computations

For $a>0$, find the minimum value of $\int_0^1 \frac{ax^2+(a^2+2a)x+2a^2-2a+4}{(x+a)(x+2)}dx.$ 2010 Gakusyuin University entrance exam/Science

2010 Today's Calculation Of Integral, 541

Tags: calculus , integration , function , calculus computations

Find the functions $ f(x),\ g(x)$ satisfying the following equations. (1) $ f'(x) \equal{} 2f(x) \plus{} 10,\ f(0) \equal{} 0$ (2) $ \int_0^x u^3g(u)du \equal{} x^4 \plus{} g(x)$

2011 Today's Calculation Of Integral, 738

Tags: calculus , integration , trigonometry , calculus computations

Answer the following questions: (1) Find the value of $a$ for which $S=\int_{-\pi}^{\pi} (x-a\sin 3x)^2dx$ is minimized, then find the minimum value. (2) Find the vlues of $p,\ q$ for which $T=\int_{-\pi}^{\pi} (\sin 3x-px-qx^2)^2dx$ is minimized, then find the minimum value.

2009 ISI B.Stat Entrance Exam, 5

Tags: calculus , rectangle , calculus computations , geometry

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

2009 Today's Calculation Of Integral, 418

Tags: calculus , integration , calculus computations

(1) 2009 Kansai University entrance exam Calculate $ \int \frac{e^{\minus{}2x}}{1\plus{}e^{\minus{}x}}\ dx$. (2) 2009 Rikkyo University entrance exam/Science Evaluate $ \int_0^ 1 \frac{2x^3}{1\plus{}x^2}\ dx$.

2013 Today's Calculation Of Integral, 875

Tags: calculus , integration , trigonometry , ratio , calculus computations

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$