**1.773 volume questions.**

Imagine, I choose $a$ as the volume of the cone. Find an expression with $a$ as variabele for the minimum surface.
My Work:
$$\text{Volume}=\frac{\pi r^2h}{3}=a\to r=\sqrt{\frac{3a}{\pi h}}$$
$$\...

I was doing a question Magical Box on geeksforgeeks, here the question asks to find the maximum volume of cuboid with given perimeter and area, so for this i found its formula on https://stackoverflow....

Let K be a solid figure in $\mathbb{R^3}$, $\mu$ the 3-dimensional Lebesgue-measure and suppose we know, that $\int_{\mathbb{R}^n} f(2*x_1,...,2*x_n) d\mu_n=\frac{1}{2^n}\int f(y_1,...,y_n)d\mu_n$ (...

So how do i find the volume of an ellipse rotated about the y=x.
Thanks.

Swimming pool is of length 20m, wide 5m and height of the swimming pool increase from the 1.6m to 4.4m.
What is the volume of swimming pool?
How I approached:
Area of swimming pool = Area of cube + ...

Does the formula $V = 1/3 \cdot S\cdot h$ also apply to:
Skewed (oblique) cone with any-shaped base - including elliptic base and any other figure imaginable base (which area is $S$) and height (...

I am required to find the volume common to ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2} = 1$ and cylinder $x^2 + y^2 = ay$
I set up the following triple integral:
$$V=2\cdot \int_{}^...

Find the volume bounded by the $xy$ plane, cylinder $x^2 + y^2 = 1$ and sphere $x^2 + y^2 +z^2 = 4$.
I am struggling with setting up the bounds of integration.
First, I will calculate the 'first-...

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids?
Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid $$E(c,A)=\{x|(...

I'm about to enter graduate school and I'm preparing for a placement exam involving some advanced calculus. I found this problem on one of the past exams and I've been stuck on it for awhile.
Find ...

I have a polytope $P$ described as the convex hull of finite points $u_1,..., u_m\in \mathbb R^n$. Is there an easy way to compute the volume of $P$ in $\mathbb R^n$?
So far I have it written as
$$
...

Consider the following set (spectrahedron/spectraheplex)
$$\mathcal A = \{ W : W \succeq 0, \mbox{tr}(W)=1 \}$$
Consider an approximating set
$$\mathcal B = \mbox{co} \{ u_i u_i^T : \|u_i\|_2 = 1, ...

I would really appretiate a clear explanation.
Solve using lagrange functio.Which body, bounded by the planes $x = 0, y = 0, z = 0$ and the plane tangent to the
ellipsoid
$$\frac{x^2}{a^2} + \frac{y^...

I'm really sorry, this may sound ridiculous but I can't understand the Wikipedia explanation about the volume of regular n-dimensional simplices, here.
In particular, these two sentences make no ...

Given is the following inertia tensor of a certain mass distribution $\rho(\vec{r})$ :
$$ I_{ij} = \int dV \rho(\vec{r}) \left( \vec{r}^2 \cdot \delta_{ij} - r_ir_j \right) $$
I should compute the ...

I was looking to formulate a general solution for a vertical slice of a cone. After failing to do this by integrating for the area of an ever shrinking chord segment, I eventually came up with a more ...

I have a capsule shape that is comprised of a cylinder and two half-sphere end caps, and I want to dynamically resize it. As I stretch the capsule by increasing/decreasing the height of the cylinder, ...

What is the volume of the solid generated when the region bounded by $y=3x$ and $y=x^2 +2$ rotate about the $x$-axis ?

Problem statement (see pic below):
1. Wrap unit circle over cylinder with radius $r=1/\pi$, edges of unit circle will just touch each other.
2. Remove cylinder, fill cannoli with cream)).
3. Scrape ...

Suppose we have a cube, and I want to integrate the function $f(x,y,z)$ over it where $x,y,z$ are the variable in given three dimensions. Assuming cube's center is at a distance of $d$ from the point ...

Let's say that three consecutive edges of a parallelepiped be a , b , c . Then how to show that volume is = [a b c]
Or = a.(b × c) ?
Also how to prove this this too?

Supposing we have a point $(x_0,y_0, z_0)$ on a plane ${x^a+y^a+z^a=1}$ where $x,y$ and $z$ are positive, and a plane P which is a tangent to that point. And we need to calculate the volume of the ...

The region $W$ is the cone shown below (see image).
The angle at the vertex is $π/3$, and the top is flat and at a height of $7\sqrt{3}$.
Write the limits of integration for $\int_W dV$ in the ...

Given is a Paraboloid delimited as following:
$$z_1 = a(x^2 + y^2),\ z_2 = h $$
That's my try for the Volume computation:
First I find the radius of the circle resulting from the intersection ...

T is shaped like a quarter of a donut, and can be described in cylindrical coordinates by the inequality (r−2)^2 + z^2 ≤ 1. 0≤θ≤π/2. I am asked to find the volum of T. I think you can parameterize it ...

I recently arrived to the chapter that focuses on finding volumes with integrals in the textbook I use. Integration is not the problem, but I've been having a very hard time solving these kind of ...

Find the volume of the solid formed by rotating the region enclosed by the curves $y=(e^ x) + 2$, $y=0$ , $x=0$, and $x=0.7$ about the $x$-axis
I set up the equations as follows using the washer ...

I'm trying to find the volume of a region contained within a sphere centered at the origin of radius 2, and above the plane z=1. I made the computation below and just wanted another pair of eyes to ...

I am very inexperienced when it comes to math and I'd like to calculate the area of an irregular semi circle. This shape is 200 ft wide and an average of 20 feet deep. (Like a wide trough shape). Any ...

I have a three-dimensional matrix (representing a volume) with integer values in it (grey values). I now want to put a (maybe oblique) plane inside the volume and extract the values that project onto ...

I know that the volume of an icosahedron is $\frac {5(3+\sqrt{5})} {12}x^3$, where x is the length of any given side. However, I am not sure how to prove it. I have looked into a few methods, yet none ...

Volume form on a manifold is alternating and this lead to impossibility on integration on a non-orientable manifolds (because there is no continuous valume form on them)
But if we consider absolute ...

I've been given the graph y = $4x - 4x^2$ that has been bound by the x-axis. I've been asked to find the volume of Solid of Revolution about the y-axis using the disc method.
In order to find the ...

Find the volume of of the wedge shaped solid that lies above the xy plane, below the $z=x$ plane and within the cylinder $x^2+y^2 = 4$.
I'm having serious trouble picturing this. I think the z ...

Given two cuboids in 3D space (8 vertex coordinates each) that are arbitrarily oriented, how can we find the volume of overlap between them in the fastest manner? An algorithm that one can code up ...

The Plane: $x+y=4$
The Cylinder: $y^2+9z^2=16$
I have gone this far but I'm not sure it's true
$$V=\int\limits_0^4\int\limits_0^{4-x}\int\limits_0^{\sqrt{16-y^2}/9} dz\ dy\ dx$$
PS: Answer can ...

I found the following problem in a Math Olympiad book:
I know we could find the volume considering that the figure is symmetric and using a solid of revolution and an integral, the problem is that ...

- integration
- calculus
- geometry
- multivariable-calculus
- definite-integrals
- area
- spheres
- 3d
- solid-of-revolution
- linear-algebra
- solid-geometry
- optimization
- spherical-coordinates
- surfaces
- real-analysis
- measure-theory
- multiple-integral
- algebra-precalculus
- differential-geometry
- rotations
- determinant
- polar-coordinates
- derivatives
- analysis
- vectors

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