Articles of volume

Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$

Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$ Here is an image: Normally I can do this question, but this one is tricky because since we are rotating about the y-axis, and we are quadratic, when I solve for $x$ I get two answers, one positive and […]

How deep is the booze in my cocktail glass?

A cocktail glass is typically filled with 130 milliliters of liquid. My cocktail glass has a diameter of 115 millimeters. If I stick a needle in the middle of my filled cocktail glass, the booze would reach a height of 37.55 millimeters (ignoring the volume of the needle completely). If I take another, identical cocktail […]

Probability of spherical particles touching one another in a cylindrical column

I was wondering if you could help me out with a little problem, as my research is getting rather time restricted at the moment, and due to my limited mathematical background, any help would be greatly appreciated, even if it was just to point me in the right direction. I have an experiment that involves […]

Find the volume between two paraboloids

Find the volume of the solid enclosed by the paraboloids $z = 1-x^2-y^2$ and $z = -1 + (x-1)^2 + y^2$. Using triple integrals, it is known that $V = \iiint_R \mathrm dx\,\mathrm dy\,\mathrm dz$, and I will have to change variables. But I can’t just say that $r^2 = x^2 + y^2$, because the […]

Question about volume preserving transformations

I will denote by $d$-vol the $d$ dimensional volume in $\mathbb{R}^D$, where $D \geq d$. For example, if $A=\{(x,y,0)\in\mathbb{R}^3:\text{max}(|x|,|y|)\leq 1\}$, then $3\mbox{-}\text{vol}(A)=0$ but $2\mbox{-}\text{vol}(A)=4$. I know that if $f:\mathbb{R}^D \rightarrow \mathbb{R}^D$ is a differentiable and invertible function, the change of $D$-volume induced by $f$ is $|\text{det}(J)|$, where $J$ is the Jacobian of $f$. Thus, for […]

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ submanifold. Then the Hausdorff dimension of $\Sigma$ is $n$ and, for any relatively open set $U \subset \Sigma$ $$\mathcal{H}^n(U)= \int_U \mathrm{dvol}$$ where $\mathrm{dvol}$ denotes the usual volume […]

Calculate volume of intersection of 2 spheres

I need calculate volume of intersection(common part) 2 spheres using integrals. First sphere has center [5,0,0] and radius = 5 second sphere has [15,0,0] and radius = 8 I wrote functions $$f_1(x)=\sqrt{5^2-(x-5)^2}$$ $$f_2(x)=\sqrt{8^2-(x-15)^2}$$ I found intersection $$f_1(x)=f_2(x)$$ $$x=\frac{161}{20}$$ I need find volume using integral, but I am not sure if I am using good formula: […]

Find number of Cuboids in a larger Cuboid

This is a question posed to my brother in Grade 5. What would be the general approach to solve- How many cuboids of dimensions $a*b*c$ are there in a cuboid of dimension $d*e*f$? My brothers approach: Just get the ratio $\frac{d*e*f}{a*b*c}$ and round it off. My approach: Check 1: Each of $d$, $e$, $f$ must […]

What is the size of $\{(x_1,\ldots,x_n)\in\mathbb{R}^n:x_1+\cdots+x_n<a\text{, and }x_i>0\}$?

This question already has an answer here: Volume of $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$ 4 answers

Proportion of cube closer to centre than outside

The question ‘what proportion $p_2$ of a square is closer to the centre than the outside?’ has been asked here before and the answer shown to be $\frac{4\sqrt{2}-5}{3}$. In another question asking the same for an equilateral triangle an answer is given generalizing this to a regular $n$-gon. I once extended the solution method of […]