5 votes
Accepted

Calculate the improper integral $\int_{B(\mathbf{0}, 1)} \frac{\mathrm{d} x \mathrm{~d} y \mathrm{~d} z}{1-a x-b y-c z}$

As the domain as well as the measure of the integral $$I(\vec{n})=\!\!\!\int\limits_{B(0, 1)}\! \! d^3x \; \frac{1}{1- \vec{n}\cdot \vec{x}}, \qquad |\vec{n}|=1,$$ are invariant under rotations, we ...
Hyperon's user avatar
  • 883
2 votes

Integrate a sum of trig function under absolute value

First, some symmetry considerations. Since $\cos x$ takes on the same values $[0,\pi]$ as it does on $[\pi,2\pi]$, we have that $$\int_{[0,2\pi]^n}\cdots d^nx = 2^n\int_{[0,\pi]^n}\cdots d^nx$$ We ...
Ninad Munshi's user avatar
  • 34.5k
2 votes

How to calculate a multiple integral over a triangular region

You can do it using polar coordinates. In fact,\begin{align}\int_0^1\int_0^{1-x}e^{\frac12(x+y)^2}\,\mathrm dy\,\mathrm dx&=\int_0^{\pi/2}\int_0^{1/(\cos(\theta)+\sin(\theta))}\rho e^{\frac12\rho^...
José Carlos Santos's user avatar
2 votes
Accepted

How to calculate a multiple integral over a triangular region

You are quite close - it's a good thought! Continuing the geometric intuition - you want to represent the points on the line $x+y =u$ with positive $x$ and $y$ coordinates. For positive $u$, observe ...
Dhanvi Sreenivasan's user avatar
1 vote

Double Integral Problem in Polar Coordinates

$$\int_0^R \frac{r}{\sqrt{r^2+2 r R \cos (\phi )+R^2}} \, dr=R \left(\sqrt{2} \sqrt{\cos (\phi )+1}-\cos (\phi ) \log \left(\sqrt{\sec ^2\left(\frac{\phi }{2}\right)}+1\right)-1\right)$$ $$\int_0^{2 \...
gpmath's user avatar
  • 956
1 vote

set the limits of integration of the spherical coordinates between two paraboloids and a plane

Let's firstly consider our figure only in first quadrant and at the end will multiply result on $4$. Let me consider spherical coordinates $$\begin{cases}x=r\sin\phi\cos\theta & \\ y= r \sin\phi\...
zkutch's user avatar
  • 13.4k
1 vote
Accepted

set the limits of integration of the spherical coordinates between two paraboloids and a plane

There are two components of the integral that must be considered: When $\rho$ varies between the inner and outer paraboloid When $\rho$ varies between the inner paraboloid and the plane $z=1$. We ...
whpowell96's user avatar
  • 5,595
1 vote
Accepted

Calculate the volume of solid

Yes, the answer is $47\pi$. This is so because the volume of the cylinder is$$\int_0^{2\pi}\int_0^2\int_0^2\rho\,\mathrm d\rho\,\mathrm dz\,\mathrm d\theta=8\pi,$$whereas the volume of the cone is$$\...
José Carlos Santos's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible