Finding C from a definite Integral

[dBanji]

Hi everyone, it's me again.

I just need an idea or call it a nudge in the right direction regarding finding a value for C such that adding C to the antiderivative recovers the definite integral ${\displaystyle \int }$ f(x) dx from a to x. I am a bit confused about what approach to use. It appears to me like Fundamental Theorem of Calculus part 2 but it's just not adding up. I have the derivative and the integral but to get the C is the problem.

The problem is solve for antiderivative f of f with C=0. ${\displaystyle \int }$ 1/${\displaystyle \sqrt{}}$(9-x^2) [-3,3]. I got the antiderivative as sin^-1(${\displaystyle \frac{x}{3}}$)
The other question is the problem right now i.e. find a value for C such that adding C to the antiderivative recovers the definite integral \"int" \ f(x) dx from a to x

I will appreciate an idea.
I hope the questions are clear enough.
Thank you.
PS: I really have not done anything meaningful on the second part of the question so nothing copy and attach.

 

[Dr.Peterson]

Hi everyone, it's me again.

I just need an idea or call it a nudge in the right direction regarding finding a value for C such that adding C to the antiderivative recovers the definite integral ${\displaystyle \int }$ f(x) dx from a to x. I am a bit confused about what approach to use. It appears to me like Fundamental Theorem of Calculus part 2 but it's just not adding up. I have the derivative and the integral but to get the C is the problem.

The problem is solve for antiderivative f of f with C=0. ${\displaystyle \int }$ 1/${\displaystyle \sqrt{}}$(9-x^2) [-3,3]. I got the antiderivative as sin^-1(${\displaystyle \frac{x}{3}}$)
The other question is the problem right now i.e. find a value for C such that adding C to the antiderivative recovers the definite integral \"int" \ f(x) dx from a to x

I will appreciate an idea.
I hope the questions are clear enough.
Thank you.
PS: I really have not done anything meaningful on the second part of the question so nothing copy and attach.

Please show us the exact wording of the problem, ideally in the form of an image. Also, please don't make things so hard to read by mixing in bold and italic. I might have an answer for you if I weren't so dizzy.

Have you tried just doing the definite integral to see what it looks like?

 

[nasi112]

I think that you have a misunderstanding about find C for a definite integral. In definite integrals C gets cancelled. In other words, definite integrals does not have C.

 

[Dr.Peterson]

I think that you have a misunderstanding about find C for a definite integral. In definite integrals C gets cancelled. In other words, definite integrals does not have C.

No, I think the problem is asking for a value of C, but the problem is more interesting than you think! I just want to make sure I'm interpreting it correctly.

The definite integral shown will be a function of x (the upper limit), and therefore will in fact be a particular antiderivative, with a particular value for C. And the FTC (in one of its forms) will be one way to explain the answer.

 

[nasi112]

[MATH]\int_{a}^{x} f(x) \ dx = F(x) + C - F(a) - C = F(x) - F(a)[/MATH]
This is what I understand. I also prefer to use a different differential element than [MATH]dx[/MATH] since the limit contains the variable [MATH]x[/MATH]
I would do it like this,

[MATH]\int_{a}^{x} f(t) \ dt = F(x) + C - F(a) - C = F(x) - F(a)[/MATH]
Let me wait and see how you will interpret the problem and what does the OP want exactly. It would be so interesting if it means something else than what I did.

 

[Dr.Peterson]

Observe the similarity between "F(x) + C" and "F(x) - F(a)". I think that is related to what they want. (I'm intentionally stating that just a little obliquely.)

And, yes, I would use dt, too; that may even be a miscopied part of the problem, though it can be interpreted validly with the dx.

@dBanji, does our discussion help you see what to try? My expectation is that there will be two ways to think about the problem, which will instructively give the same result. But I'm waiting to see the exact problem, and your attempt, so we have more to talk about.

 

[dBanji]

Observe the similarity between "F(x) + C" and "F(x) - F(a)". I think that is related to what they want. (I'm intentionally stating that just a little obliquely.)

And, yes, I would use dt, too; that may even be a miscopied part of the problem, though it can be interpreted validly with the dx.

@dBanji, does our discussion help you see what to try? My expectation is that there will be two ways to think about the problem, which will instructively give the same result. But I'm waiting to see the exact problem, and your attempt, so we have more to talk about.

Sorry, I slept off as I got tired of trying to figure it out lol. The question is correct. I have attached a shot of the question exactly. I hope this gives a better understanding of the question.

The question uses ${\displaystyle dx}$ and I thought ${\displaystyle dt}$ would be better but that was how it was framed. I will take a closer look at it now that I just woke up but if you can further explain a better approach, that will be great.

Sorry about the bold characters I thought I was using it to bring out salient points since I still need to figure out how to properly type maths characters.
Thank you.

 

[dBanji]

I think that you have a misunderstanding about find C for a definite integral. In definite integrals C gets cancelled. In other words, definite integrals does not have C.

Thank you for replying. Yes, dinite integrals do not need a C but this is asking for one lol. I was a little confused too as this is the first time I was seeing a question like that. I am not sure if to use the limit of integration [-3,3] given in the question. When I did I got ${\displaystyle \pi }$ as the answer but that was not correct. I tried to take the derivative of both the integral and the antiderivative but got lost completely

 

[dBanji]

I have tried again and got pi as the answer. Where am I getting it wrong? See attached my hand-written work.

 

[Dr.Peterson]

I have tried again and got pi as the answer. Where am I getting it wrong? See attached my hand-written work.

You seem to be taking the definite integral from -3 to 3, rather than from a constant a to the variable x. Where they mention [-3,3], I believe that is just stating the domain.

Also, the antiderivative is just your F(x):


Find that definite integral, and show us what you get. You should see what C is, when you compare this to your antiderivative.

 

[dBanji]

You seem to be taking the definite integral from -3 to 3, rather than from a constant a to the variable x. Where they mention [-3,3], I believe that is just stating the domain.

Also, the antiderivative is just your F(x):
View attachment 26706

Find that definite integral, and show us what you get. You should see what C is when you compare this to your antiderivative.

Thanks for responding.

I'm not sure I understand the "find that definite integral" part of your statement. Did you mean that I should find the definite integral of ${\displaystyle \int }$ f(x)dx with regards to x and the constant C?

 

[Dr.Peterson]

I'm not sure I understand the "find that definite integral" part of your statement. Did you mean that I should find the definite integral of ${\displaystyle \int }$ f(x)dx with regards to x and the constant C?

They ask about . Evaluate it!!!

On the other hand, I just realized they never actually define what they mean by f(x). I'm assuming they mean the integrand, so that what they are asking for is





Do it.

I will not be online much of today, so I'll make my next observation now:

You found what F(x) is; you want to add to that a C such that when x = a, you will get 0. Do you see why? This is presumably what they are really asking you to do, which will match what you get when you do the definite integral.

 

[dBanji]

They ask about View attachment 26709 . Evaluate it!!!

On the other hand, I just realized they never actually define what they mean by f(x). I'm assuming they mean the integrand, so that what they are asking for is


View attachment 26710


Do it.

I will not be online much of today, so I'll make my next observation now:

You found what F(x) is; you want to add to that a C such that when x = a, you will get 0. Do you see why? This is presumably what they are really asking you to do, which will match what you get when you do the definite integral.

Thanks.
Hmm, I'll try and see. So the constant C should be a number not an expression right?

 

[dBanji]

They ask about View attachment 26709 . Evaluate it!!!

On the other hand, I just realized they never actually define what they mean by f(x). I'm assuming they mean the integrand, so that what they are asking for is


View attachment 26710


Do it.

I will not be online much of today, so I'll make my next observation now:

You found what F(x) is; you want to add to that a C such that when x = a, you will get 0. Do you see why? This is presumably what they are really asking you to do, which will match what you get when you do the definite integral.

The antiderivative of the function is sin^-1(${\displaystyle \frac{x}{3}}$) + C. If evaluated from ${\displaystyle a}$ to ${\displaystyle x}$ and added to C (when ${\displaystyle x}$=${\displaystyle a}$), the result is simply 0. I still do not get the part thay says "such that when x=a you will get a 0). This C is stuborn.

 

[dBanji]

They ask about View attachment 26709 . Evaluate it!!!

On the other hand, I just realized they never actually define what they mean by f(x). I'm assuming they mean the integrand, so that what they are asking for is


View attachment 26710


Do it.

I will not be online much of today, so I'll make my next observation now:

You found what F(x) is; you want to add to that a C such that when x = a, you will get 0. Do you see why? This is presumably what they are really asking you to do, which will match what you get when you do the definite integral.

What I understand from what you explained is this:

The antiderivative F(x) = sin^-1(${\displaystyle \frac{x}{3}}$). such that when you add C to that i.e. sin ^-1(${\displaystyle \frac{x}{3}}$) + C, you will get a 0 provided x=a. How to make that happen is now the challenge, i.e. evaluating the from a to a plus C =0. I still cannot see how sin ^-1(${\displaystyle \frac{a}{3}}$) can be evaluated

 

[Dr.Peterson]

If a is not given, then all you can get is an expression! So what expression represents C? You almost have it!

 

[dBanji]

If a is not given, then all you can get is an expression! So what expression represents C? You almost have it!

Thank you for getting back. What I have as my result is F(x) - arcsin(${\displaystyle \frac{a}{3}}$). This means that the expression that represents C is arcsin(${\displaystyle \frac{a}{3}}$).
What do you think?

 

[Dr.Peterson]

Thank you for getting back. What I have as my result is F(x) - arcsin(${\displaystyle \frac{a}{3}}$). This means that the expression that represents C is arcsin(${\displaystyle \frac{a}{3}}$).
What do you think?

I would replace F(x) with the expression for it (since the question didn't define that), and you're good.

 

[dBanji]

I would replace F(x) with the expression for it (since the question didn't define that), and you're good.

Thank you very much for your help and time.