Double integral involving greatest integer function and change of variables

In order to evaluate $\displaystyle\int_{0}^{2}\int_{0}^{2}\lfloor x+y \rfloor dx \,dy$, I transformed co-ordinates using $x= u-v$ and $y=v$. Then the Jacobian becomes $1$ and $0 \leq u \leq 4$ and $0 \leq v \leq 2$ implying that$\displaystyle\int_{0}^{2}\int_{0}^{2}\lfloor x+y \rfloor dx \,dy = \displaystyle\int_{0}^{2}\int_{0}^{4}\lfloor u \rfloor du\,dv = \displaystyle\int_{0}^{2}6\, dv = 12$However the answer on Wolfram alpha is $6$. Can someone point out where I went wrong?...Read more

multiple integral - Computational Challenge of Multivariate Integration in Sympy

I am creating a piecewise linear approximation for the following equation:$$W = \frac{\theta \left(k \rho\right)^{k}}{2 k^{2}k! \rho \left( 1- \rho\right)^{2} \left(\frac{\left(k \rho\right)^{k}}{\left(1- \rho\right) k!} + \sum_{n=0}^{k - 1} \frac{\left(k \rho\right)^{n}}{n!}\right)}$$as $W_{approx}=A+B\rho+C\theta+Dk$. In this equation, $\theta$, $\rho$, and $k$ are the variables, where $\theta,\rho\in \mathbb{R}^+$, $k\in \mathbb{Z}^{>0}$, and $0\leq\rho<1$. To do that multivariate linear regression, I want to minimize the norm:$$F = \i...Read more

multiple integral - Do we have to take the absolute value of the jacobian ONLY if it is a number?

if we want to evaluate the integration$$I=\int\int(x^3y^3)(x^2+y^2)dA$$over the region bounded by the curves$$xy=1,\\xy=3,\\x^2-y^2=1,\\x^2-y^2=4$$I used the transformation $$u=xy,\\v=x^2-y^2$$I found that the jacobian will be $$J=\frac{1}{-2y^2-2x^2}$$Do I have to get the absolute value of the jacobian?If I did not take the absolute value , I will get the result of the integration = - 30if I take the absolute value$$J=\frac{1}{2y^2+2x^2}$$I will get the result= + 30 .My friend told me we take absolute value of the jacobian only if it is a numb...Read more

Explicit Formula for Multiple Integral

$$ \int_0^1 \! \int_0^1 ...\int_0^1 \frac{dx_1dx_2...dx_n}{(a+x_1+x_2+...+x_n)^m}$$ I've tried for n=2, m=1, I got : $a\ln a - 2(a+1)\ln (a+1)+(a+2)\ln(a+2)$n=2, m=2, I got : $-\ln a + 2\ln(a+1)-\ln(a+2)$n=3, m=1, I got : $\frac{1}{2}(-a^2\ln a+3(a+1)^2\ln (a+1)-3(a+2)^2\ln(a+2)+(a+3)^2\ln (a+3))$but I still have no idea what's next. Or there is a tricky way??...Read more

How to solve double integral using integration by parts?

How to solve following double integral using the integration by parts?\begin{equation} \int_{0}^{T} \bigg\{\big[\sin{(i+j)\tau}-\sin{(i-j)\tau}\big]\int_{0}^{\tau}\frac{\cos{js}}{s^p}ds\bigg\}d\tau \end{equation}The solution looks something like this:\begin{equation} \bigg[\frac{\cos{(i+j)\tau}}{(i+j)}-\frac{\cos{(i-j)\tau}}{(i-j)}\bigg]\int_{0}^{\tau}\frac{\cos{js}}{s^p}ds\bigg|_{0}^{T}-\int_{0}^{T}\bigg[\frac{\cos{(i+j)\tau}}{(i+j)}-\frac{\cos{(i-j)\tau}}{(i-j)}\bigg]\frac{\cos{j\tau}}{\tau^p}d\tau\end{equation}How to get it?...Read more

multiple integral - To find volume using triple integration

Find volume of the solid above XY plane and directly below the portion of the elliptic paraboloid $x^2+\frac{y^2}{4}=z$ which is cutoff by plane $z=9$Now I came up with $\int \int x^2 + \frac{y^2}{4} $ dxdy. After using change of variable as $x = 3rcos\theta$ and $y =6rsin\theta $. I got the integral as$\int_{0}^{2\pi} \int_{0}^{1} 9r^2 \cdot18 \cdot r drd\theta$ = $81 \pi$Please check if this is correct or not ?Thanks...Read more

Jacobian Transformations (Multiple Intergrals)

I have a question: Evaluate $\iint_{D} xy(\sqrt{1-x-y})\,dx\,dy$, where $D$ is the region bounded by $x=0$, $y=0$, $x+y=1$ using the transformations $x+y=u$ and $y=uv$.I can see that the region $D$ is a triangle with coordinates $(0,0)$, $(1,0)$ and $(0,1)$. However in my book when transformed the new region is a rectangle with the vertices $(0,0)$, $(1,0)$, $(0,1)$ and $(1,1)$.My question: I am unable to understand how the new region would turn out to be rectangle. How does the new region turn out as a rectangle?For $(0,0)$, we have $u=0$ and ...Read more

What is meant by the path an integral integrates over?

I have some questions about what the path an integral integrates over actually means.I can one understand one interpretation of what is meant by $L$ in the following line integral.$$ \Delta U = -\int_{\vec{s} \in L} \vec{F} \cdot \mathrm{d} \vec{s} $$Clearly, $L$ is not a set of points because than the integral giving the expression for moving up from the surface of the earth to a point in the sky would be the same as moving from the point in the sky to the surface of the earth (instead of being the negation.)One can rephrase the expression in ...Read more

Limt of multiple integral

Let $~f:\mathbb{R}\to\mathbb{R}$ be continuous non-negative function and $\int_{-\infty}^{+\infty} f(x)dx = 1$Denote : $$I_n(r) = {\idotsint}_{x_1^2+\dots x_n^2 \le r^2} \prod_{k = 1}^{n} f(x_k)dx_1\cdots dx_n$$How to find the following limit : $\lim_{n\to\infty} I_n(r)$ for fixed r....Read more

Calculate a multiple integral

Calculate $\iint|cos(x+y)|dxdy$ where $0\leq x \leq π$ and $0\leq y \leq π$ using the transformation $x=u-v$ and $y=v$ This question is bothering me, as I am having problems regarding changing the region of integration, changing the limits. It would be better if full solution is provided....Read more

Help calculating integral over a region $G$ with spherical coordinates?

So i have to calculate this triple integral:$$\iiint_GzdV$$Where G is defined as: $$x^2+y^2-z^2 \geq 6R^2, x^2+y^2+z^2\leq12R^2, z\geq0$$So with drawing it it gives this:It seems i should use spherical coordinates since sphere is involved.But i don't know if i can use the default ones or should manipulate them since i have hyperbole of some sort(but since its $a=b=c$, is it a double sided cone then? i know that z>0 means im only looking at upper half, but still)So using this:$$x=r\cos\phi\cos\theta; y=r\sin\phi\cos\theta ; z=r\sin\theta$$So i t...Read more

My question is about the integration limits of a double integral

I have to solve the following exercise :Find the area of the $R$ region that is enclosed by the parable $y=x^2$ and the line $y=x+2$. I did the following shape:And I solved the double integral:(the integration limits to $x$ axe: $y=x^2$ to $y=x+2$ and $-1\leq x\leq 2$ )$A= \int_{R}^{}\int_{}{}dA=\int_{-1}^{2}\int_{x^2}^{x+2}dydx=\int_{-1}^{2}[y]_{x^2}^{x+2}dx=\int_{-1}^{2}(x+2-x^2)dx=[\frac{x^2}{2}]_{-1}^{2}+[2x]_{-1}^{2}-[\frac{x^3}{3}]_{-1}^{2}=\frac{9}{2} $If we reverse the sequence of integration (from $dydx$ to $dxdy$), I am confused about...Read more

multiple integral - Find the volume of the solid bounded by the xy plane, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=4$.

Find the volume of the solid bounded by the xy plane, the cylinder $x^{2} + y^{2}=4$, and the plane $z+y=4$.If we draw the graph, then the integral will be calculated should be$$ \int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}} (4-y) \: dy dx $$This would give the volume of the solid in 1st quadrant which can also be obtained through $$ \int_{0}^{2} \int_{0}^{\sqrt{4-y^{2}}} (4-y) \: dx dy $$which would be equal to $4\pi - \frac83$. Both the above equations give the same result.But if we try to find the volume of the entire solid formed by the three curv...Read more