# Finding Volume and Surface Area of a Banana Using Calculus

• TKDonuts
In summary, the conversation revolved around using calculus to find the volume and surface area of a banana. The participants had learned elementary calculus and volumes of revolutions. They traced the banana on graph paper and used a water displacement method to measure its actual volume. They attempted to find the volume using an integral and the radius using the formula for the area of a circle, but encountered issues with their equations. They also mentioned needing to find a method for calculating the surface area of the banana. According to their equations and assuming a circular cross section, the volume of the banana was estimated to be about 286 cubic units.
TKDonuts

## Homework Statement

We are given a Banana, and asked to find the volume and surface area of the function, using calculus. So far, we have learned elementary calculus (derivatives, limits, and integrals) as well as volumes of revolutions. We traced the banana on graph paper, plotted points on the line, and created equations for the cross section of the banana. Here is the graph on desmos: https://www.desmos.com/calculator/jfertkhwnv
We measured the actual volume of the banana, using a water displacement method. The actual volume of the banana is 160cm^3

## Homework Equations

We are allowed to use any credible resource for equations online.

## The Attempt at a Solution

So far, what we have tried is to find the integral between the two lines we traces, (the integral from one end to the other end of the banana, of f(x)-g(x).)(couldn't figure out how to put integral sign in thing.) We then multiplied that value by 1/2, to find the approximate radius. then the calculated radius was plugged into pi*r^2 equation, and we got a value that was incredibly high. Perhaps our equations were incorrect, but at this point, we do not know. We also need to find a method to find the surface area of the banana, however we have not started that phase of the project.

For what it's worth, using your equations and assuming circular cross section for the banana I get a volume of a bit over ##286## cubic units.

## 1. How is calculus used to find the volume and surface area of a banana?

Calculus is used to find the volume and surface area of a banana by breaking the banana's shape into smaller, simpler shapes, such as cylinders and spheres. These shapes can then be analyzed using calculus equations and techniques to determine the total volume and surface area of the banana.

## 2. What information is needed to find the volume and surface area of a banana using calculus?

To find the volume and surface area of a banana using calculus, you will need to know the length, width, and height of the banana. In addition, you will also need to know the radius of any curved sections of the banana, such as the curved sides or the ends.

## 3. What is the difference between volume and surface area?

Volume refers to the amount of space inside a three-dimensional object, while surface area is the measurement of the total area of the object's outer surface. In the case of a banana, the volume would be the amount of space inside the banana, while the surface area would be the measurement of the banana's skin.

## 4. Can calculus be used to find the volume and surface area of any object?

Yes, calculus can be used to find the volume and surface area of any object that has a defined shape and dimensions. By breaking the object down into smaller, more manageable shapes and applying calculus equations, the volume and surface area of the object can be determined.

## 5. Why is it important to find the volume and surface area of a banana using calculus?

Finding the volume and surface area of a banana using calculus can be useful for a variety of purposes, such as in engineering and design. It can also help in understanding the properties and characteristics of the banana, which can have implications in fields such as nutrition and agriculture.

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