Mathematics Stack Exchange
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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

401 questions
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Solution volume of solid of revolution

Find the solid volume formed by the rotation of the region bounded by $x = 0, x = 1, y = 0, y = 5 + x ^ 6$ on the $x-\text{axis}.$ I make: $\int_0^1 \pi \cdot ((5+x^6)-0)^2$. And find: $(\cfrac{1}{13} +\cfrac{10}{7} + 25) \cdot \pi.$
robert
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Solid of revolution for area bounded by parabolas

Show that the volume generated by revolving the region in the first quadrant bounded by the parabolas $y^{2} =x$, $y^{2}= 8x, x^{2}= y, x^{2}= 8y$ about the x axis is $279 \pi /2$ This problem is to be solved by changing the variables as $ y^{2} =…
Mathaddict
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Volume of solid by revolution

Find the volume generated by revolving the area bounded by the curves $(x^2+4a^2)y=8a^3, 2y=x$ and $x=0$, about the y axis. To find volume I am using this formula: $V=\pi\int [f(y)]^2dy$ where $f(y)= \sqrt{\frac{8a^3}{y}-4a^2}$ But how do I…
user464068
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bezier surface patch for surface of revolution

I have the following control points for 2 bezier surface patches trying to build a surface of revolution out of them. float ctlpoints[][4] ={{-2,-1, 0, 1}, {0, 0, 2, 0}, {2,-1, 0, 1}, {-3, 0, 0, 1}, {0, 0, 3, 0}, {3, 0, 0,…
Hisham Sueyllam
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Solid of revolution for given area

Find the volume of the solid obtained by revolving the area enclosed by the curve $27ay^{2} = 4(x-2a)^{3} , x$ axis and parabola $y^{2} = 4ax$ about the $x$ axis. I am not able to find the area enclosed by both curves. I know that $27ay^{2} =…
Mathaddict
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Volume of the solid generated by revolution of the given curve.

The volume obtained on revolving about $x=a/2$, the area enclosed between the curves $xy^{2} = a^{2}(a-x)$ and $(a-x)y^{2} = a^{2}x$ is ......$?$ I've drawn both curves and both intersect at $x=a/2$, but the line $x=a/2$ lies in middle of both…
Mathaddict
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Why is the slope a part of the area of a surface of revolution?

I have previously learned about the volume of a solid of revolution about the x axis and that equation makes sense to me since it's taking the integral between points a and b of the area of a circle at each point. $$ \int_{a}^{b} \pi [f(x)]²dx =…
Jesper
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How to find the coefficients of this Integral

I am encountering a very tricky problem for me and I am not sure the right approach to solve it. It is simple. It is just telling me to fill in the coefficients of the integral to find the volume of the figure. But I honestly do not how to approach…
Niko H
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Help understanding solution in volume of Solid of Revolution $y=12-x^3$ and $y=12-4x$ revolved around 1.) x-axis 2.) y-axis

The curves look like this. The solution for the volume used a circular disk for the x axis of revolution. I got the total volume from doubling the volume by rotating the solid formed from $x=0$ to $x=2$. Trying to apply the same concept from the…
Renzo
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Revolution about $x$-axis Error

Can someone please explain to me where my error is in this problem because i should not be getting a negative volume. Find the volume of the solid generating by revolving this region about the x-axis $$y= 1 - |x|, y=0. $$ I have attached my work,…
Hritik
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Volume by revolution of quadrant of circle about its tangent

An area bounded by a quadrant of a circle of radius $a$ and the tangents at its extremities revolves about one of the tangents. Find the volume so generated. I am trying to use this formula: $V=\pi\int_0^a [f(y)]^2dy$ I have used…
user464068
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