Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [solid-of-revolution]

This tag is for questions regarding to "Solid of revolution", a three-dimensional object obtained by rotating a function in the plane about a line in the plane.

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis of revolution) that lies on the same plane.

The following figure gives a clear concept about it.

enter image description here

Notes:

  • If the curve was a circle, we would obtain the surface of a sphere.
  • If the curve was a straight line through the origin, we would obtain the surface of a cone.
  • A representative disk is a three-dimensional volume element of a solid of revolution.

Reference:

https://en.wikipedia.org/wiki/Solid_of_revolution

436 questions
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Surface Area vs. Volume of Solid of Revolution

Surface Area and Volume of Solid of Revolution: Why does $\int 2 \times \pi y \, dx$ not work for surface area, but $\int \pi \times y^2 \, dx$ works for volume? I know that for surface area, it’s because the function is slanted so it couldn’t be…
user335699
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3 answers

Minimizing surface area of revolution for a fixed volume of revolution

Suppose I want an open-topped cup to have the capacity to hold a volume $V$ of liquid. I want to find a shape for my cup that minimises its surface area. My attempt I strongly suspect that the shape of this cup will be rotationally symmetric,…
Ultra
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Why this solids also lives below z axis?

The base of a certain solid is the circle $x^2 + y^2 = a^2$. Each plane perpendicular to the x-axis intersects the solid in a square cross-section with one side in the base of the solid. Find its volume. Here is my try : $$y = \sqrt{a^2-x^2}$$ $$dV…
SirMrpirateroberts
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Finding the volume using Washers

Problem: Find the volume generated when the region bounded by the given curves and line is revolved about the x-axis. $$ y = 3x - x^2$$ $$ y = 3x $$ Answer: Let $V$ be the volume we are trying to find. The first step is to find the points where $3x…
Bob
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Disc vs Shell Method, getting different answers AP calc

Can someone please check my work. $R$ is the region in the first quadrant bounded by $y=1/x$, $y=1$ and $x=e$ Find the volume of the solid generated when $R$ is revolved about the line $y=1$ Disk: $$V= \int_{1}^{e} \pi \left(1-\frac{1}{x}\right)^2…
user130306
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1 answer

Find surface which generated by revolving a line in $\mathbb{R}^3$

Problem : Let $l$ be a line which passes two points : $(1,0,0), (1,1,1)$. And $S $ be a surface which generated by revolving line $l$ around $z$-axis. Find a volume enclosed by surface $S$ and two planes : $z=0, z=1$. My Attempt Parametric…
bFur4list
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4 answers

Why do we care so much about solids/surfaces of revolution?

Problems involving solids/surfaces of revolution seem to be a fairly standard part of any calculus curriculum (at least in the USA), but the topic is so incredibly specific that I can't think of any motivation that doesn't involve pottery wheels or…
R. Burton
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Pappus centroid theorem and Hypercones.

The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have: $$ V_{C3}=\int_0^h \pi r^2 dz=\pi \int_0^h \frac{R^2}{h^2}z^2 dz=\pi…
Emilio Novati
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Equal area parameterization of a torus?

I am trying to parameterise a surface of revolution such that each infinitesimal area element is uniform across the surface. The cross-sections of the surface are shown in the picture below. The title of the post refers to a torus as I thought this…
Peanutlex
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2 answers

Evaluating $\int_1^e{\sqrt{\ln x}}dx$ by finding volume

$$\int_1^e\sqrt{\ln x}\;\mathrm{d}x$$ WolframAlpha provides an answer to the integral in terms of the imaginary error function. However, I was wondering why the method I employed did not work: I can construct a solid formed by rotating $\sqrt{\ln…
sreysus
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Volume obtained by rotating the area enclosed by $y=x^2$ and $y=\sqrt x$ about $y=1$ vs about $x$-axis

When I scatched the area, I expected the volume obtained by rotating about $y=1$ to be identical with the volume obtained by rotating about the $x$-axis. To my surprise, calculation shows different results. According my calculation, volume with…
user138819
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Verifying formulas and process for surface area and volume of a spindle torus

While working on this geometry problem I reasoned that the surface area of the spindle torus is the surface area of the apple (outer surface) plus the surface area of the lemon (inner surface) while the volume of the spindle torus is the volume of…
user1167128
4
votes
1 answer

Volumes of Rotation: Where's my mistake??

I have tried to calculate the volume of of the solid bound by the function $y = (-x^2+6x-5)^2$ and $y = x-1$ around the line $x =6$. Here is my work. The answer should be $\frac{63\pi}{2}$ but I'm getting a different answer. Have I made a mistake…
Adam Wiltzer
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4 answers

Find a solid of revolution whose volume is 72π and whose surface area is 36π.

I have tried setting up multiple systems of equations using many known volumes but I always seem to come up short. My last attempt was a hollow cylinder but that leaves you with three unknowns in only two sim. equations (for V and S.A). Can anyone…
mbstackbm
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4
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3 answers

What is the correct formula for the washer method?

Almost everywhere I look the formula is: $$ \pi \int_b^a {\left(f(x)^2 - g(x)^2\right) dx} $$ where f(x) is the big function and g(x) is the smaller function. Though I've run into problems while calculating the volume using this formula, and through…
Jackson
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