Finding the integral of an unknown function

In summary, the bonus problem asks for the value of the integral of f'(x)dx from 0 to 1. The only assumption necessary is that f is continuous.
  • #1
pattiecake
64
0
I need the help of all you math folks out there...I'm working on a bonus assignment in my calculus 2 class. Here's the problem:

Suppose the curve y=f(x) passes through the origin and the point (1,1). Find the value of the integral of f'(x) dx from 0 to 1.

I thought this question just wanted a restatement of the fundamental theorem of calculus. But since it's a bonus problem, I know that there has to be something tricky here. It does ask for a "value".

Obviously this curve can take any shape, so there's no general formula for the area.

Does anyone have any suggestions?
 
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  • #2
Hi pattie,
Write down what you know about f mathematically, and write down your integral. The deceptively simple result is just one of the many results that illustrate the power of calculus.
The only prerequisite assumption missing from your statement is that f(x) is continuous.
 
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  • #3
The book doesn't even specify it's continuous...but assuming it is I have the integral of f'(x)dx from 0 to 1 is [f(1)-f(0)]. Is it really that simple?
 
  • #4
Have you proven/read a proof of the fundamental theorem of calculus ? :smile: Unless you spot a flaw in the proof, it really is that simple.
 
  • #5
well i guess you can assume that f' is integrable, since they said find its integral. then the MVT implies that an integrable function which is a derivative, can be comouted as you say, i.e. the FTC holds. continuity is not needed.

this stronger proof of FTC, without continuity, like most other things i know about calc can be found in courant.
 
  • #6
If they say to compute the integral of f', it's obvious to assume f is differentiable and hence that f is continuous.
 
  • #7
The continuity of f isn't the issue. It is the continuity of df/dx that is "missing".
 

What is the process for finding the integral of an unknown function?

The process for finding the integral of an unknown function involves using various techniques such as substitution, integration by parts, and partial fraction decomposition to manipulate the function into a form that can be integrated. Once this is achieved, the integral can be solved using the fundamental theorem of calculus.

Do I need to know the function in order to find its integral?

No, you do not need to know the function in order to find its integral. The techniques used for integration do not require knowledge of the specific function, only the form in which it needs to be manipulated in order to be integrated.

Can I use a calculator or computer to find the integral of an unknown function?

Yes, you can use a calculator or computer to find the integral of an unknown function. There are many online tools and software programs available that can solve integrals for you, but it is still important to have a basic understanding of the integration process.

How do I know if I have correctly found the integral of an unknown function?

You can check if you have correctly found the integral of an unknown function by taking the derivative of the integral and seeing if it matches the original function. If the two functions are equivalent, then you have correctly found the integral.

What are some real-world applications of finding the integral of an unknown function?

Finding the integral of an unknown function has many real-world applications, such as in physics, engineering, economics, and statistics. It can be used to calculate areas, volumes, and rates of change, among other things. For example, in physics, integrals are used to calculate work, energy, and displacement over a given time period.

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