Articles of volume

How to find maximum and minimum volumes of solid obtained by rotating $y=\sin x$ around $y=c$

I would appreciate if somebody could help me with the following problem: Q: Let $S$ be the region bounded by the curves $y=\sin x \ (0 \leq x \leq \pi)$ and $y=0$. Let $V(c)$ be the volume of the solid of obtained by rotating the region $S$ around the line $y=c \ (0 \leq c […]

Volume of the largest rectangular parallelepiped inscribed in an ellipsoid

Show that the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ is $\dfrac{8abc}{3\sqrt3}$. I proceeded by assuming that the volume is $xyz$ and used a Lagrange multiplier to start with $$xyz+\lambda \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}-1\right)$$ I proceeded further to arrive at $\frac{abc}{3\sqrt3}$. Somehow I seemed to be have missed $8$. Can someone […]

Volume of a truncated conical frustum

If you have a conical frustum with a volume $$V_f=\frac{\pi h}{3}\left( r^2+rR+R^2 \right)$$ with $h$ being the distance between the bases, $r$ the radius of the smaller circle, and $R$ the radius of the larger. Then you truncate the frustum by a plane that intersects the larger base at one point at an angle $\theta$. […]

Volume using Triple Integrals

Find the volume of solid enclosed by surfaces $x^2+y^2=9$ and $x^2+z^2=9$ I understand that these are two cylinders in XY and XZ planes respectively, that will cut each other above the XY plane. I get the following limits for triple integrals $$\int_{\theta=0}^{2\pi}\int_{r=0}^3\int_{z=0}^{\sqrt{ 9-r^2\cos^2\theta}}rdzdrd\theta$$ Is it correct ?? If yes, the integral itself looks so complicates. […]

Spherical Cake and the egg slicer

Recently I baked a spherical cake (3cm radius) and invited over a few friends, 6 of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took my egg slicer with equally spaced wedges(and designed to cut 6 […]

How should this volume integral be set up?

I would like to find the (four-dimensional) volume of the region given by $$xy>zw \quad\wedge \quad x>-y \quad\wedge \quad x^2+y^2+z^2+w^2<1,$$ for $x,y,z,w\in\mathbb{R}$ and where the last condition means that the whole thing is bounded by the unit $4$-ball. The boundary of the region given by the two first conditions would be (I think) \begin{align} xy=zw […]

Is it possible to create a volumetric object which has a circle, a square and an equilateral triangle as orthogonal profiles?

This question was posed to me by a friend (formulated as creating a peg to fit perfectly into holes of these shapes), and after an experiment in OpenSCAD it seems it is not possible – either one profile has to be an isosceles triangle rather than equilateral, rectangular rather than square, or elliptical rather than […]

Volume of irregular solid

I need to calculate volume of irregular solid which is having fix $200 \times 200$ width and breadth but all four points varies in depth. I have table which gives depth at each point. How to calculate volume of such solid? Hi, I am giving here my main problem definition. I have a grid with […]

Maximum volume change for two sets with small Hausdorff metric in bounded part of $\mathbb{R}^n$

Given two subsets $S_1$, $S_2$ of a bounded part of $\mathbb{R}^n$, say $[-M,M]^n$. Is there a way to relate the difference in volume $vol(S_2)-vol(S_1)$ to the Hausdorff metric distance between the sets $S_1$ and $S_2$ given that $S_1 \subset S_2$? Intuitively I can see that the the fact that $S_1$ and $S_2$ lie in $[-M,M]^n$ […]

Volume of the projection of the unit cube on a hyperplane

Let $C_n\subset\mathbb{R}^n$ be the $n$-dimensional cube with side $1$, and let $P_k$ be any $k$-dimensional plane, $k\leq n$. What is the maximal $k$-volume $V_{n,k}$ of the projection of $C_n$ on $P_k$? Quite obviously, the minimal area should be $1$, obtained by taking $C_n = [0,1]^n$ and projecting it on $\{\mathbf{x}\in\mathbb{R}^n|x_{k+1}=\ldots=x_n=0\}$. I think the maximum should […]