# Calculus NoviceEvaluating Washer/Shell Volumes - Calculus Novice

• CalcNoob
In summary, the conversation discusses a question about evaluating volumes of shapes with no solid interiors, specifically regarding determining the outer and inner functions. The speaker also mentions difficulties in determining the functions when no graph is provided and suggests using absolute value to account for potential errors.
CalcNoob
Hi there,

I'm new here and I'm really glad that I found a community to discuss calculus even though its not really my favorite course (although I enjoy some of its topics).

My first question here is about evaluating volumes of shapes that have no solid interiors. I know how to use the washer formula, my problem is determining the outer function and the inner function. I know I'm supposed to subtract the inner shape volume from the outer one. But in some problems, they give us the functions in which the region is enclosed by (e.g. f(x) and g(x), y=a,y=b )but they don't provide any graph for it. How am I supposed to know if f(x) > g(x) for all [a,b] or vice-verse ?
Usually, in the exams they give us known functions but with modifications , such as y=sqrt(x^2-25), I think such functions are easier to graph it. Thanks and advance, and sorry for my bad english , hopefully you understood my question :-)

Regards,

I'm not sure it will matter very much, if you get a negative value by using choose the wrong functions for inner/outter, then just absolute value it. Same problem as not knowing wether a or b is bigger to choose a-b or b-a, absolute valuing it will give you the difference anyway, which is what you want, The difference between the outer volume and inner volume. Hope i helped.

Calculus Novice

Hi Calculus Novice,

Welcome to the community! I'm glad you found a place to discuss calculus and I hope you'll find it helpful and enjoyable.

To answer your question about evaluating volumes of shapes that have no solid interiors, it's important to first understand the concept of the washer formula. The washer formula is used to find the volume of a solid formed by rotating a region between two curves about a horizontal or vertical axis. The outer function is the curve farthest from the axis of rotation, while the inner function is the curve closest to the axis of rotation. The volume is then calculated by subtracting the volume of the inner shape from the volume of the outer shape.

In cases where the functions are given but no graph is provided, you can use algebraic techniques to determine which function is the outer and inner function. For example, if y=f(x) is the upper curve and y=g(x) is the lower curve, you can set the two functions equal to each other and solve for the points of intersection. The function with the larger value at the points of intersection will be the outer function.

In cases where the functions are given with modifications, such as y=sqrt(x^2-25), you can use the same techniques to determine the outer and inner functions. You can also use a graphing calculator or online graphing tool to plot the functions and visually see which one is the outer and inner function.

I hope this helps answer your question. Don't worry about your English, it's perfectly understandable. Keep practicing and you'll continue to improve. Best of luck with your calculus studies!

Regards,

## 1. What is the purpose of evaluating washer/shell volumes in calculus?

Evaluating washer/shell volumes is an important concept in calculus because it allows us to find the volume of irregular shapes that cannot be easily calculated using traditional methods. This technique is particularly useful in real-world applications such as engineering and physics, where finding the volume of complex objects is necessary.

## 2. How do you determine the volume of a washer or shell using calculus?

To find the volume of a washer, you first need to calculate the area of the base of the washer by subtracting the smaller circle's area from the larger circle's area. Then, you multiply this area by the height of the washer to get the volume. For a shell, you calculate the volume by finding the circumference of the base circle and multiplying it by the height of the shell.

## 3. What is the difference between a washer and a shell in calculus?

A washer is a solid shape formed by rotating a region between two curves around an axis, while a shell is a solid shape formed by rotating a region bounded by a curve and a line around an axis. In simpler terms, a washer has two curves while a shell has one curve and one line.

## 4. Can washer/shell volumes be negative?

No, washer/shell volumes cannot be negative. Since volume is a measure of space and cannot have a negative value, the volume of a washer or shell will always be positive.

## 5. What are some real-life examples where evaluating washer/shell volumes is useful?

Evaluating washer/shell volumes is used in various fields such as architecture, engineering, and physics. Some examples include calculating the volume of a water tank, finding the volume of a car engine's pistons, and determining the volume of a chemical reaction vessel in a laboratory.

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