Found problems: 1687
2010 Today's Calculation Of Integral, 585
Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.
2012 Today's Calculation Of Integral, 816
Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.
2011 Today's Calculation Of Integral, 720
Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.
2010 Today's Calculation Of Integral, 617
Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane.
Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$
[i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]
2010 Today's Calculation Of Integral, 595
Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$
2009 Kumamoto University entrance exam/Medicine
2011 Today's Calculation Of Integral, 678
Evaluate
\[\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).\]
[i]2011 Doshisya University entrance exam/Life Medical Sciences[/i]
1985 AMC 12/AHSME, 21
How many integers $ x$ satisfy the equation
\[ (x^2 \minus{} x \minus{} 1)^{x \plus{} 2} \equal{} 1
\]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{none of these}$
2010 Today's Calculation Of Integral, 658
Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$
(1) Find the length $L$ of $C$.
(2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$.
Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School.
[i]1997 Kyoto University entrance exam/Science[/i]
2019 Jozsef Wildt International Math Competition, W. 31
Let $a, b \in \Gamma$, $a < b$ and the differentiable function $f : [a, b] \to \Gamma$, such that $f (a) = a$ and $f (b) = b$. Prove that $$\int \limits_{a}^{b} \left(f'(x)\right)^2dx \geq b-a$$
2005 Today's Calculation Of Integral, 63
For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$
Evaluate
\[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]
2010 Today's Calculation Of Integral, 621
Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$
[i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]
2000 Moldova Team Selection Test, 3
For each positive integer $ n$, evaluate the sum
\[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]
2010 Today's Calculation Of Integral, 555
For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.
2010 Today's Calculation Of Integral, 560
Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$.
(1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis.
(2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis.
Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.
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 $\dfrac {\sqrt 3} 2$.
2013 Today's Calculation Of Integral, 864
Let $m,\ n$ be positive integer such that $2\leq m<n$.
(1) Prove the inequality as follows.
\[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\]
(2) Prove the inequality as follows.
\[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\]
(3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows.
\[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]
2013 Today's Calculation Of Integral, 862
Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$
2007 Today's Calculation Of Integral, 224
Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.
2015 District Olympiad, 2
[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $
[b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $
[i]Florin Stănescu[/i]
Today's calculation of integrals, 868
In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation.
(1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$.
(2) Find the volume of the common part of $V_1$ and $V_2$.
2010 Today's Calculation Of Integral, 573
Find the area of the figure bounded by three curves
$ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$
$ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$
$ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.
2009 Today's Calculation Of Integral, 514
Prove the following inequalities:
(1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$
(2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$
2009 Today's Calculation Of Integral, 481
For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$.
Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$
1948 Putnam, B4
For what $\lambda$ does the equation
$$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$
have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?
2009 Today's Calculation Of Integral, 482
Let $ n$ be natural number. Find the limit value of ${ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).$