Found problems: 713
2007 Today's Calculation Of Integral, 255
Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.
2011 Today's Calculation Of Integral, 757
Evaluate
\[\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.\]
2007 Today's Calculation Of Integral, 195
Find continuous functions $x(t),\ y(t)$ such that
$\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$
$\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$
2013 Today's Calculation Of Integral, 885
Find the infinite integrals as follows.
(1) 2013 Hiroshima City University entrance exam/Informatic Science
$\int \frac{x^2}{2-x^2}dx$
(2) 2013 Kanseigakuin University entrance exam/Science and Technology
$\int x^4\ln x\ dx$
(3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam
$\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$
Today's calculation of integrals, 877
Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$
Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$
2011 Today's Calculation Of Integral, 755
Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane.
Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.
2007 Today's Calculation Of Integral, 184
(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality.
\[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \]
(2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$
2011 Today's Calculation Of Integral, 684
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]
2005 Today's Calculation Of Integral, 22
Evaluate
\[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]
2010 ISI B.Math Entrance Exam, 2
In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$.
[asy]
Label f;
xaxis(0,60,blue);
yaxis(0,60,blue);
real f(real x)
{
return (x^2)/60;
}
draw(graph(f,0,53),red);
label("$y=x^2$",(30,15),E);
real f(real x)
{
return (x^2)/25;
}
draw(graph(f,0,38),red);
label("$y=2x^2$",(37,37^2/25),E);
real f(real x)
{
return (x^2)/10;
}
draw(graph(f,0,25),red);
label("$y=f(x)$",(24,576/10),W);
label("$O(0,0)$",(0,0),S);
dot((20,400/25));
dot((20,400/60));
label("$P$",(20,400/25),E);
label("$B$",(20,400/60),SE);
dot(((4000/25)^(0.5),400/25));
label("$A$",((4000/25)^(0.5),400/25),W);
draw((20,400/25)..((4000/25)^(0.5),400/25));
draw((20,400/25)..(20,400/60));
[/asy]
2010 Today's Calculation Of Integral, 550
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.
2005 Today's Calculation Of Integral, 44
Evaluate
\[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]
2009 Today's Calculation Of Integral, 433
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {(\sin x)^{\cos x}}{(\cos x)^{\sin x} \plus{} (\sin x)^{\cos x}} dx$.
2010 Today's Calculation Of Integral, 548
For $ f(x)\equal{}e^{\frac{x}{2}}\cos \frac{x}{2}$, evaluate $ \sum_{n\equal{}0}^{\infty} \int_{\minus{}\pi}^{\pi}f(x)f(x\minus{}2n\pi)dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
2009 Today's Calculation Of Integral, 502
(1) For $ 0 < x < 1$, prove that $ (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}.$
(2) Find $ \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}$.
2006 ISI B.Stat Entrance Exam, 1
If the normal to the curve $x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23}$ at some point makes an angle $\theta$ with the $X$-axis, show that the equation of the normal is
\[y\cos\theta-x\sin\theta=a\cos 2\theta\]
2010 Today's Calculation Of Integral, 655
Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.
2009 ISI B.Stat Entrance Exam, 5
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.
2010 Today's Calculation Of Integral, 619
Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized.
Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$
[i]2010 Saitama University entrance exam/Mathematics[/i]
Last Edited
Today's calculation of integrals, 894
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$
2010 Today's Calculation Of Integral, 608
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, 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.
Today's calculation of integrals, 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 $F(t)\geq 1.$
Today's calculation of integrals, 866
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions.
(1) Find the cross-sectional area $S(x)$ at the hight $x$.
(2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$
2005 Today's Calculation Of Integral, 2
Calculate the following indefinite integrals.
[1] $\int \cos \left(2x-\frac{\pi}{3}\right)dx$
[2]$\int \frac{dx}{\cos ^2 (3x+4)}$
[3]$\int (x-1)\sqrt[3]{x-2}dx$
[4]$\int x\cdot 3^{x^2+1}dx$
[5]$\int \frac{dx}{\sqrt{1-x}}dx$