Who is right, me or my professor? proof area of a circle

[Rafeeg]

Homework Statement

"Set up an integral involving a function and evaluate the integral to prove the formula for the area of a circle of radius r is pi*r^2. Show all steps."

2. The attempt at a solution

I imagined the circle as an infinite tiny arc lengths or "circumference's", with each arc length = 2*pi*R, where R is changing as we move from the center to the outer ring and dR as thickness

so

Area = integration of (2*pi*R*dR) from 0 to r
which gives pi*r^2

My professor does not like this answer, she said our lesson is about calculus arc length and I have to use techniques from that lesson. But I don't find anything in the question requesting that.

Also would be kind if you can link the proof from other websites so I can show her, because she doesn't trust students.

Regards
Rafeeg

 

[shelovesmath]

Homework Statement

"Set up an integral involving a function and evaluate the integral to prove the formula for the area of a circle of radius r is pi*r^2. Show all steps."




2. The attempt at a solution

I imagined the circle as an infinite tiny arc lengths or "circumference's", with each arc length = 2*pi*R, where R is changing as we move from the center to the outer ring and dR as thickness

so

Area = integration of (2*pi*R*dR) from 0 to r
which gives pi*r^2


My professor does not like this answer, she said our lesson is about calculus arc length and I have to use techniques from that lesson. But I don't find anything in the question requesting that.

Also would be kind if you can link the proof from other websites so I can show her, because she doesn't trust students.

Regards
Rafeeg

You probably can't win either way. If you're wrong, you're wrong. If you're right, then you're possibly making her feel stupid or just plain annoying her. It's a shame your professor isn't fostering more of an atmosphere of intellectual curiosity even if your answer is completely wrong. Although I don't have an actual answer for you myself, I would encourage you to keep thinking about math in a curious fashion regardless of whether your professor approves or not. I believe this is where math becomes fun.

 

[Rafeeg]

shelovesmath, I wish that was the case, the problem is, this is a graded exam question and I want my grades back if I'm correct.

 

[Hootenanny]

Welcome to Physics Forums.

I will not criticize a teacher. However, you method is a classical approach to computing the area of a circle and is the shell method in 2D, sometimes called the onion method. Knowing the arc length of a circle, you can compute the area by simply "summing" the contributions from all circles with radii [itex]0\leq R \leq r[/itex], as you did.

Perhaps, your teacher is looking for something further. For example, how would you proceed if you did not know the circumference of a circle?

 

[Rafeeg]

Thanks Hootenanny
So that means I am correct since the question did not specify the method needed.
I have searched for the onion method and shells method but did not find something relevant to this approach, would appreciate a website that I could use to show my professor.

 

[Hootenanny]

Thanks Hootenanny
So that means I am correct since the question did not specify the method needed.
I have searched for the onion method and shells method but did not find something relevant to this approach, would appreciate a website that I could use to show my professor.

Search for the area of a disc and you should find plenty of references. Wikipedia for example, has a good page on the area of a circle.

The final decision on whether you get the marks or not rests with your Professor. Are you sure that the question wasn't in a section related to "arc lengths" or something similiar, or the question didn't say "using the concept of arc length..."?

 

[Rafeeg]

Yup, I'm sure the question was stated generally without anything stating or relating to the method of approach needed.
The part from Wikipedia is exactly what I need

Wish you all the best

 

[shelovesmath]

Yes, if she wasn't explicit in the method to use, as long as the answer is correct, you should be fine. Otherwise, take it up with your school's math chair. If, however, it turns out you misread the directions, get ready to put your foot in your mouth.

 

[vela]

Going to the math chair over a couple of points on an exam is an overreaction, and the department will simply tell you it's the instructor's class and she gets to evaluate students' work the way she wants. All instructors have very likely dealt with students complaining about their grades, so unless the error is egregious, you're really not going to find sympathy from the instructor's colleagues. The only thing you'll probably accomplish is to annoy the teacher.

It helps to keep in mind that the point of an exam isn't simply to show off that you know a way to solve a problem. It's to demonstrate that you have mastered the concepts being taught in class. There's quite often different ways to solve a problem, and you can't reasonably expect every problem to explicitly state what method should be used. To claim that you can use any method you want since the teacher didn't say you couldn't is disingenuous.

On a related note, part of the difficulty in a math course is understanding what you may assume to be true and what you can't. Rafeeg's work is fine if you can assume that the circumference of the circle is given by [itex]2\pi r[/itex], but if you're learning about how to calculate arc lengths, it's also reasonable that the instructor expects you to derive this result first if you want to use it.