# Formula for a Non-uniform Cylinder

• Eduardo Pinto
In summary, to calculate a non-standard cylinder with a varying top and base radius, you can use calculus and the concept of volumes of revolution. You can also find formulas for the volume of a frustum of a cone. Another approach is to treat it as a cone minus a smaller cone and use the formulas for cone calculations. A visual and intuitive approach can also be helpful.
Eduardo Pinto
Mentor note: moved from non homework thread to here hence no template.

Hi, I wanted to calculate a non standard Cylinder. It have a Height (h), with a base radius of (r1) and a top radius (r2). It varies linearly along the height. How can I come up with a formula for that? Since my girlfriend was Intrigued, I also wanted but I'm not sure how to come up with a formula for that.

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Eduardo Pinto said:
Hi, I wanted to calculate a non standard Cylinder. It have a Height (h), with a base radius of (r1) and a top radius (r2). It varies linearly along the height.
Sounds like you're asking about what is called a frustum of a cone. There are formulas for the volume of such a figure. See https://en.wikipedia.org/wiki/Frustum

Calculate what? The volume? The surface? The center of mass? The moment of inertia? Something else?

In many cases you can treat it as cone minus a smaller cone.

As an aside, here's a video on computing the surface area of a sphere:

If you ponder your problem in a similar way you might come up with a visual and intuitive way to solve your problem.

If not you can always use @mfb cone strategy and find the related formulas here:

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

## What is a non-uniform cylinder?

A non-uniform cylinder is a three-dimensional shape that has a circular base and varying cross-sectional areas along its height. This means that the radius or diameter of the cylinder changes as you move along its length.

## How is the volume of a non-uniform cylinder calculated?

The volume of a non-uniform cylinder is calculated by dividing it into smaller, uniform cylinders and adding up their individual volumes. This can be done using calculus, by integrating the cross-sectional area function along the height of the cylinder.

## What is the formula for the surface area of a non-uniform cylinder?

The formula for the surface area of a non-uniform cylinder is a bit more complex than that of a regular cylinder. It involves calculating the surface area of each individual section of the cylinder and adding them together. This can also be done using calculus, by integrating the lateral surface area function along the height of the cylinder.

## What are some real-life applications of non-uniform cylinders?

Non-uniform cylinders can be found in many everyday objects such as bottles, vases, and even some types of pipes. They are also commonly used in engineering and architecture for creating complex structures and designs.

## What challenges do scientists face when working with non-uniform cylinders?

One of the main challenges when working with non-uniform cylinders is accurately calculating their volume and surface area. This requires advanced mathematical skills and can be time-consuming. Additionally, the irregular shape of non-uniform cylinders can make it difficult to fit them into certain spaces or use them in certain applications.

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