Questions tagged [cross-sections]

A cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections.

Filter by
Sorted by
Tagged with
1 vote
0 answers
21 views

Find volume of solid generated by curves in $xy$ plane and cross sections perpendicular to $x$-axis

The curves are $y=x^2, y=x$ and the cross sections are squares perpendicular to the $x-$axis such that the base of the squares is on the $xy-$plane. My solution: The area of the squares is given by $A=...
user avatar
  • 3,097
8 votes
1 answer
116 views

How to find the $2d$ cross-sections of the $n$-hypercube that are regular $2n$-gons?

I'm currently trying to do something that sounds doable, but on which I've now been stuck for a while. The idea is that, if you consider an $n$-hypercube, the polygon found by doing a $2d$ cross-...
user avatar
  • 83
2 votes
2 answers
57 views

Equation of a section plane in hyperbolic paraboloid

Find the equation of a plane passing through $Ox$ and intersecting a hyperbolic paraboloid $\frac{x^2}{p}-\frac{y^2}{q}=2z$ $(p>0, q>0)$ along a hyperbola with equal semi-axes. My attempt: The ...
user avatar
0 votes
0 answers
26 views

Continuity from Group Action and Section

Assume that $\Psi: G \times Y \to Y$ is a topological group action (so $\Psi$ is continuous). Let $[\cdot]: Y \to Y_{/G}$ be the continuous projection map, sending each $y\in Y$ to its orbit. Also, ...
user avatar
  • 364
0 votes
0 answers
61 views

Projection of the oblate spheroid at a given rotation onto a slanted plane

I'm currently looking for a projection of (1) the oblate spheroid onto (2) the slanted plane. (1) Having the oblate spheroid: $\frac {x^2}{a^2}+\frac {y^2}{b^2}+\frac {z^2}{c^2}=1$ with $a=b>c$ and ...
user avatar
0 votes
2 answers
46 views

find the volume of the solid of intersection of the two spheres of radii a and b (with b $<$ a)

Also the center of the smaller one lies on the surface of the larger one. This is what I have thus far. Utilize the Cartesian plane we have for the smaller and larger spheres respectively: $(x-a)^2 + ...
user avatar
  • 325
1 vote
0 answers
28 views

Find the formula for a function $f(x,y)$ with two given cross sections

I have done problems where I am given a contour map and I am asked to find a formula of a linear function but I am having trouble with this one. I found the change in $x$ over the change in $z$ but I ...
user avatar
0 votes
1 answer
41 views

sketch of a solid for calculating volume using slicing method

In reference to the textbook: Thomas’ Calculus, 14th edition, Heil, Weir & Hass In chapter 6, section 1, Exercise 1: We need to find the volume of a solid using the slicing method It says that the ...
user avatar
1 vote
1 answer
659 views

Let $R$ be a region bounded by $y=x^2$, $y=2x$, and $y=3$. Solid $S$ is obtained by rotating region $R$ about $x=-1$.

Let $R$ be a region bounded by $y=x^2$, $y=2x$, and $y=3$. Solid $S$ is obtained by rotating region $R$ about $x=-1$. UPDATE: the region R is the region bounded below the line y=3. Write the integral ...
user avatar
0 votes
1 answer
2k views

Volumes with Cross Sections, Circle Base and Square Cross Section

The base of a solid is a circle centered at the origin with a radius r, and every plane section perpendicular to a diameter is a square. What is the volume of the solid? So this would usually be easy ...
user avatar
3 votes
1 answer
201 views

Can the cross section of parallelepiped be a regular pentagon

Came across this question in a children's recreational mathematics book. Apparently, the cross section of a cube cannot be a regular pentagon. It could be a irregular pentagon though. But if we ...
user avatar
  • 235
1 vote
1 answer
186 views

Pre-Calculus Question: How to find the line of sight that someone cannot see?

If Dave is standing next to a silo of cross-sectional radius r=9 feet at the indicated position, his vision will be partially obstructed. Find the portion of the y-axis that Dave cannot see. (Hint: ...
user avatar
0 votes
1 answer
247 views

Finding volume of solid by cross sections?

Find the volume of the solid whose base is the region bounded by 𝑦 = 𝑥² and the line 𝑦 = 1, and whose cross sections perpendicular to the base and parallel to the x-axis are squares. I'm not ...
user avatar
1 vote
1 answer
86 views

Find axial intercepts of a perpendicular plane of vector [xyz], the plane passing through its position [x,y,z] [closed]

I tried to make the question as brief as possible in the title without omitting critical aspects. There are three elements to it. 1) A position vector [xyz] is given first and then 2) Its ...
user avatar
  • 111
1 vote
2 answers
102 views

Volume of a solid with base bounded by a parabola and a straight line

The base of a solid is the region bounded by $y=0,25x^2$ and $y=3-x$. Cross sections perpendicular to the $y$ axis are rectangles with base in the XY plane and its height is twice its length. Find the ...
user avatar
  • 341
2 votes
1 answer
71 views

Vertical cross-section of $S : r(u,v) = (u+v,uv,u^2v)$.

Exercise : Consider the surface $S : r(u,v) = (u+v,uv,u^2v), \; (u,v) \in \mathbb R^2$. Express the vertical cross section $c$ of the surface at the point $(2,1,1)$ with direction $(2,1)$ and ...
user avatar
  • 20.8k
0 votes
1 answer
184 views

Find area of cross section of cylinder by the plane $x$

I am working on my scholarship exam practice (assume high school/pre-university math background) and I think I got half way through but I am not sure how I could continue. Let $r$ be a positive ...
user avatar
22 votes
6 answers
3k views

Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

Conic sections seem to get special attention in early math classes. My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some ...
user avatar
0 votes
1 answer
37 views

Do all functions represent a section of an n dimensional object by another object of n-1 dimension?

If $x^2 + y^2 = 4$ represents the section of a cone by a plane horizontal to the cone, a circle, and $y^2 - x^4 = 4$ represents the section of a cone by a plane vertical to the cone, a hyperbola, what ...
user avatar
  • 221
1 vote
1 answer
44 views

I have no idea how to solve this problem using areas of known cross section

The problem involving cross sections I am so confused on how to find volume using known cross sections. I've never understood it. This problem that I've encountered is very difficult, and I tried ...
user avatar
2 votes
0 answers
35 views

What is the terminology for a subset of a product of sets that is the product of its cross-sections?

Let $X$ and $Y$ be non-empty sets. For every $x \in X$, let $S_x$ be a non-empty subset of $Y$. Define $S := \prod_{x \in X}S_x$. $S$ is a subset of $Y^X$. I think I once saw a name given to this kind ...
user avatar
  • 10.4k
0 votes
1 answer
432 views

cross section of torus

Is there any particular name for the plane revolved about an external axis to form a torus? I was thinking of "cross section," but that could be taken as a vertical plane cutting the whole torus in ...
user avatar
1 vote
1 answer
116 views

In what sense are "projecting" and "taking sections" of polytopes dual operations?

It seems to be folklore that projecting and taking sections of polytopes are somehow "dual operations" (e.g. explicitly noted in the abstract of this paper, or suggested by this answer to an MO ...
user avatar
  • 28k
0 votes
1 answer
156 views

cross section for y=1

My class has just started Multivariable, and I'd just gotten back my quiz results. However, I don't understand why for B), the correct graph is apparently in the other direction I thought it would be —...
user avatar
  • 4,433
0 votes
0 answers
39 views

Cross sections of randomly distributed defects

A $10 \times 10 \times 10$ mm cube has $1000000$ 2-micron spheres randomly distributed thru out it. If a random cross-section was done. What would be the area of the cross-sectioned spheres and how ...
user avatar
0 votes
0 answers
32 views

Cylindric sections

When a cylinder intersects with a plane, what are the resultant shapes and curves? I think that the curves are hyperbola, parabola and line, and the shapes are circle, ellipse, rectangle and trapezium....
user avatar
  • 1
3 votes
1 answer
650 views

Finding the cross section of a prolate spheroid at a given rotation [theta (x,y), theta (x,z), theta (y,z)] on a central plane

I currently have an assignment in which I have to model the drag forces acting on a rugby ball as it rotates through the air. One of the variables in the drag force equation is the cross section of ...
user avatar
0 votes
1 answer
728 views

Volume of a solid formed by a triangle base with square cross sections parallel to a line

Reviewing for a test, I was given this problem. "The base of a solid is the region in the first quadrant bounded by the line x = -2y + 6 and the coordinate axes. What is the volume of the solid if ...
user avatar
  • 123
3 votes
1 answer
542 views

Cross sections of a cube

Suppose we take the set of all cross sections of a cube and construct from them a set $A$ whose elements are sets of vertices of the cube as follows. If there exists a cross section of the cube which ...
user avatar
1 vote
1 answer
3k views

Finding the volume of a solid s using cross sections

I am a given a problem that reads "The base of $S$ is a region enclosed by $y = 2-x^2$ and the $x$-axis. Cross-sections perpendicular to the $y$-axis are quarter circles." The instructions are "Find ...
user avatar
0 votes
1 answer
45 views

Finding the volume bound by the curves $y = 0 $ and $y = \sqrt x$ and $\space x = 4$ about $x = 0$

this is the picture I came up with. So I am revolving around the y-axis. Therefore, I should be using the function $x = y^2$ since that is the cross-section. Moreover, I will be integrating from $0 to ...
user avatar
2 votes
0 answers
30 views

How To Find Volume Using Integrals Under Many Curves and Shapes(Specifically a skull)

For a calculus project, I need to find the theorized volume of a shape made from curves and objects but the problem is I have so many shapes and different types of graphs that I have no clue how to ...
user avatar
  • 145
0 votes
0 answers
31 views

How to tell what sign to keep when parameterizing intersection of two surfaces

I have this exercise: The question as it is written in the book. $z = x^{2}$ and $z = 4y^{2}$ intersect in two curves, one of which goes through the point $(2, -1, 4)$. Parameterize that curve. My ...
user avatar
0 votes
1 answer
45 views

Determining the true shape of a section.

Consider a prism with a base given by $A(7,10,0) , B(8,18,0) , C(14,12,0) $ and a height of $10$.Determine the true size of the section made through the prism by a plane given with its frontal and ...
user avatar
  • 691
1 vote
2 answers
86 views

What is the ratio of the sum of surface area of cylinder that its base radius is $r$ and height is $2r$ to the sectional area parallel to the base?

What is the ratio of the sum of surface area of cylinder that its base radius is $r$ and height is $2r$ to the sectional area parallel to the base? I currently don't have any idea about the question. ...
user avatar
  • 499
0 votes
1 answer
573 views

What is the cross-sectional area and volume of this object?

All size of the object that is rectangular prism is being increased $2$ times. $a)$ What happens to the cross-sectional area? $b)$ What happens to the volume? I'm a bit confused. I want to take ...
user avatar
  • 499
0 votes
1 answer
184 views

Is this correct sytntax: the slice is a semicircle **parallel** to the x axis when perpendicular to y with triangle base along the x and y axes?

I just took an exam and the question was this: Find the volume where the base is a triangle with the vertices: (0,0),(4,0),& (0,4). The cross section is a semicircle and is perpendicular to the y-...
user avatar
  • 164
1 vote
3 answers
67 views

Volume of Shape

The base is the semicircle $$y=\sqrt{16−x^2},$$ where -4 $\le$ $x$ $\le$ 4. The cross-sections perpendicular to the $x$-axis are squares. $$\\$$ So far this is what I have: $\int (Area)\,dx$ $\...
user avatar
  • 633
0 votes
1 answer
72 views

Cross-Sections of Solids

I know already that conic sections (or conics) have been widely explored and many things about them are already known. I was wondering if this sort of exploration has taken place for any other sorts ...
user avatar
0 votes
0 answers
106 views

3D Ellipsoid Average Value Integrals

An ellipsoidal meteor is careening down from outer space. The comet's exterior takes the shape of $\frac{(x-30)^2}{36}+\frac{(y-70)^2}{9}+\frac{(z-40)^2}{25}=1$. When the meteor hits the Martian ...
user avatar
1 vote
1 answer
129 views

Why are the middle cross sections of dual polyhedrea the same?

A tetrahedron is self-dual, so it is no surprise that in both the tetrahedron and its dual the middle cross section is the same shape (a square). A cube and an octahedron are dual, and the middle ...
user avatar
  • 2,922
2 votes
2 answers
92 views

Single-Variable Integration Volume Problem. "When lengths change linearly, areas change quadratically"?

Let $C$ be a cone with base any shape (not necessarily a disk or an ellipse) having area $A$, and height $h$ (that is, the distance from the apex to the plane containing the base is $h$). Note that a ...
user avatar
  • 4,066
3 votes
0 answers
124 views

An inverse problem for cross sections

I apologize beforehand for the vague title and the length of the description I am using to setup my question; I can't seem to be more concise without sacrificing clarity. Call a region in the plane "...
user avatar
  • 316
1 vote
0 answers
33 views

Island Sketching

"The Island Euleria sank over the course of a year. A helicopter captured 11 photographs of the top of the island. It took a picture every time the island had sunk three feet, and one of the island ...
user avatar
0 votes
1 answer
85 views

2D Cross-section of data points

I have a set of points with x and y values. I would like to draw a straight line through these points and have a cross section through these points. Said another way, I would like to rotate these ...
user avatar
  • 1
1 vote
0 answers
97 views

How do I make a cross section with a parametric equation?

Making a cross section with an implicit equation is easy to do, simply set any one or more variables to zero. But, how would I do this using a parametric equation? Say we have a torus: $ x = (\cos(u) ...
user avatar
0 votes
1 answer
61 views

What's the cross-sectional area of a grating?

This is related to the electrical concept of resistance, but in the equation of resistance $$R = \frac{ρ l}{A}$$ there's the cross-sectional area $A$ of the conductor. Now if the conductor is a ...
user avatar
  • 6,822
0 votes
0 answers
360 views

Calculus II - Volumes of Solids (Disks/Washers...)

Could someone please clarify the process for finding the volume between two curves or under one curve? I understand that the volume (V) is the integral of the cross-sectional area (A), but I am a tad ...
user avatar
1 vote
0 answers
1k views

Sum of Residuals equal to Zero

It is clear that running a regression, the sum of Residuals should equal Zero. I also understand that when running a weighted regression the sum of weighted Residuals should equal Zero. My model is ...
user avatar
0 votes
1 answer
44 views

Unable to calculate integral for cross-section volume

I am trying to determine the volume of a solid by cross-sections, and I am having trouble determining the integral I should use to calculate it. Graph in question The base is bounded from $x = -8....
user avatar
  • 105