# Tag Info

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 ...
• 883

### 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 ...
• 34.5k

### 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^...
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 ...
• 5,672
1 vote

• 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 ...
• 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\...

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