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by Grigorios Kostakos
Mon Nov 27, 2017 10:42 am
Forum: Real Analysis
Topic: Sequence and limit of integral
Replies: 1
Views: 1711

Re: Sequence and limit of integral

We give a solution: The sequence of functions $g_n:[0, 1] \longrightarrow \mathbb{R}$ defined as \[g_n(x)=\begin{cases}\dfrac{x^{\frac{1}{n}}\log(1+x)}{x\,(1+x^{\frac{2}{n}})^{\frac{3}{2}}}\,,& x\in(0,1]\\ 0\,,& x=0\end{cases}\,,\quad n\in\mathbb{N}\,,\] converges pointwise to the function \[g(x)=\...
by Grigorios Kostakos
Sun Nov 26, 2017 8:02 am
Forum: Multivariate Calculus
Topic: Volume and area of a solid
Replies: 1
Views: 1440

Re: Volume and area of a solid

$D=\big\{(x,y)\in{\mathbb{R}}^2\;\big|\;(x-x_0)^2+(y-y_0)^2\leqslant d^2\big\}$ is a closed disk with center $(x_0,y_0)$ and radius $d=\sqrt{\frac{b-a}{2c}}$. The paraboloids $z=a+c(x-x_0)^2+c(y-y_0)^2$, $z=b-c(x-x_0)^2-c(y-y_0)^2$, by which the solid \[V=\Big\{(x,y,z)\in{\mathbb{R}}^3\;\big|\;(x,y)...
by Grigorios Kostakos
Thu Nov 16, 2017 4:49 pm
Forum: Multivariate Calculus
Topic: Double Integrals - Changing Order of Integration
Replies: 6
Views: 2898

Re: Double Integrals - Changing Order of Integration

Oh, I see...The example was taken from Marsden-Tromba's Vector Calculus. These books aren't the gospel truth after all! Marsden-Tromba's Vector Calculus is a good book (not "the gospel truth" for me) but even the masterpieces did not escape entirely from typos! P.S. So, I suppose that your question...
by Grigorios Kostakos
Thu Nov 16, 2017 3:59 pm
Forum: Multivariate Calculus
Topic: Double Integrals - Changing Order of Integration
Replies: 6
Views: 2898

Re: Double Integrals - Changing Order of Integration

I tried drawing D but I got confused. I represent $x=\sqrt{y}$ as $y=x^2$ and plot the lines $y=2$, $x=1$. It seems to me that D is divided in two sections: $0\leq x\leq1$ , $0\leq y\leq x^2$ and $1\leq x\leq\sqrt{2}$ , $x^2\leq y\leq 2$ , because the curve intersects the vertical line. Is this pos...
by Grigorios Kostakos
Thu Nov 16, 2017 2:28 pm
Forum: Multivariate Calculus
Topic: Double Integrals - Changing Order of Integration
Replies: 6
Views: 2898

Re: Double Integrals - Changing Order of Integration

The double integral is calculated over the closed region $D$ which can be represented as $$D=\big\{(x,y)\in\mathbb{R}\;|\; \sqrt{y}\leqslant {x}\leqslant 1, \; 0\leqslant {y}\leqslant 2\big\}\,.$$ Can you represent the same region $D$ in such way, such that the variable $x$ takes values from $0$ to ...
by Grigorios Kostakos
Wed Nov 15, 2017 3:05 pm
Forum: Multivariate Calculus
Topic: Volume between two surfaces using double/triple integrals
Replies: 2
Views: 1651

Re: Volume between two surfaces using double/triple integrals

The surface $F_1=\big\{\big(x,y,\sqrt{x^2+y^2}\,\big)\;|\; (x,y)\in\mathbb{R}^2\big\}$ is a cone and the surface $F_2=\big\{\big(x,y,2-x^2-y^2\big)\;|\; (x,y)\in\mathbb{R}^2\big\}$ is a hyperboloid. These two surfaces intersect at the circle $C=\big\{\big(x,y,1\big)\;|\; x^2+y^2=1\big\}$. (see pictu...
by Grigorios Kostakos
Wed Oct 18, 2017 11:23 am
Forum: Real Analysis
Topic: \(\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}\)
Replies: 0
Views: 1031

\(\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}\)

For the sequence $\left({\alpha_{n}}\right)_{n\in\mathbb{N}\cup\{0\}}$ of real numbers defined recursively as \[\alpha_{n}=2\,\alpha_{n-1}+2^{-2(n-1)}\,,\; n\in\mathbb{N}\,,\quad \alpha_0=1\,:\] Find the general form of \(\alpha_{n}\). Find the values of real number $\beta$ for which the \(\displays...
by Grigorios Kostakos
Wed Oct 04, 2017 1:23 pm
Forum: Calculus
Topic: \(\int_{0}^{\pi}\arcsin(1-\sin{t})\,dt\)
Replies: 0
Views: 1382

\(\int_{0}^{\pi}\arcsin(1-\sin{t})\,dt\)

Does this $$\displaystyle\int_{0}^{\pi}\arcsin(1-\sin{t})\,dt$$
can be (accurately) evaluated?
by Grigorios Kostakos
Sun Sep 24, 2017 6:37 pm
Forum: Multivariate Calculus
Topic: Volume and area of a solid
Replies: 1
Views: 1440

Volume and area of a solid

Let $a,b,c,d$ positive real numbers such that $a<b$, $d^2=\dfrac{b-a}{2c}$ and the closed disk \[D=\big\{(x,y)\in{\mathbb{R}}^2\;\big|\;(x-x_0)^2+(y-y_0)^2\leqslant d^2\big\}\,.\] Find the volume and the surfase area of the solid \[V=\Big\{(x,y,z)\in{\mathbb{R}}^3\;\big|\;(x,y)\in{D}\,,\; a+c(x-x_0)...
by Grigorios Kostakos
Tue Sep 19, 2017 7:48 pm
Forum: Multivariate Calculus
Topic: A tough limit
Replies: 1
Views: 1546

Re: A tough limit

The function $f(x,y)=x^y\,,\; (x,y)\in{\mathbb{R}}^2\,,$ is at least twice continuously differentiable in a open disk with centre $(1,1)$ and its 2nd degree Taylor polynomial is \begin{align*} P_{2,f,(1,1)}(x,y)&=f(1,1)+({\rm{grad}}\,{f})(1,1)\cdot(x-1,y-1)+\frac{1}{2}\,(x-1,y-1)\,H_f(1,1)\,(x-1,y-1...