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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.

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 —...
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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 ...
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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....
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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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-...
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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: $\int (Area)\,dx$ $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 "...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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....
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How do you check if a coordinate $(x,y)$ is inside or on the perimeter of a cross

$1.$ How do you check if a $(x,y)$ coordinate is inside a cross? $2.$ How do you check if a $(x,y)$ coordinate is on the perimeter of a cross? The cross is like a medical sign. The cross will have $...
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Volume of a polar solid of known cross section

I am trying to generate a solid of known cross sections to be semi-circles for the cardioid $r = 1 + \sin\theta$. I figured that I would need to calculate the volume of individual wedges and sum all ...
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A simple way to get the cross section of a sphere.

I'd like to cut the $3$-sphere of radius $1$ with the plane $x+y+z=k$. Of course the cross-section is a circle: the problem is that I'd like to have the equation of the circle expressed by two "...
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All cross sections of a volume

Is there a elegant or analytical way to determine all the possible shape cross section shapes of a volume given the particular volume? Example (by brute force slicing a plane through said volume in ...
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Finding a vector $\vec{a}$ that is perpendicular to a certain plane

Consider a cylinder with elliptical perpendicular cross-sections to $z-$axis; $2x^2+y^2=1$. Now I 'm asked to find a vector $\vec{a}$ being perpendicular to which the cylinder has circular cross-...
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Area of a circle from the edge to a point offset from the center

I am trying to come up with a way to calculate the cross-sectional area of the shape shown in the figure below. My first method would be to subtract the circle from the rectangle like this: $$(Y)\left(...
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How to know what type of cross section is it going to be?

A plane intersects a right rectangular pyramid. Producing a cross section. The plane is parallel to the base. What shape is the cross section? I thought it would be triangle cause triangle cut is ...
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Construct the intersection of a cube by a plane through $3$ points on its edges, no pair of which is on the same face

So this is a rather old problem, but I still cannot find a pure constructive solution to it. Please, do not offer me to write a plane equation, etc. I would be grateful, if you offer a solution by ...