How Do Zeroes of a Function Relate to Its Integral?

In summary, I think that the zeros of the original function have something to do with the function that I integrated.
  • #1
Kolika28
146
28

Homework Statement


You are given the function

f(x)=3x^2-4x-8

a) Find the values of a.
upload_2018-9-18_18-53-40.png

Explain the answers using the function.

Homework Equations

The Attempt at a Solution


a^3-2*a^2-8*a=0
a=-2 v a=0 v a=4

I found the answers, but I don't know how to explain my answers by using the function. What does the zeros of the orginal function has to do with the function I integrated?
Thank you so much for help
 

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  • #2
Kolika28 said:

Homework Statement


You are given the function

f(x)=3x^2-4x-8

a) Find the values of a.
View attachment 230840
Explain the answers using the function.

Homework Equations

The Attempt at a Solution


a^3+2*a^2+8*a=0
a=-2 v a=0 v a=4[/B]

I found the answers, but I don't know how to explain my answers by using the function. What does the zeros of the orginal function has to do with the function I integrated?
Thank you so much for help

What's the most common interpretation of an integral?
 
  • #3
Hmm, it gives you the area under the curve of a function?
 
  • #4
Kolika28 said:
Hmm, it gives you the area under the curve of a function?

It does indeed!
 
  • #5
I'm sorry, but I still don't see the importance of the zeros . I'm a slow learner
 
  • #6
Kolika28 said:
I'm sorry, but I still don't see the importance of the zeros . I'm a slow learner

Draw a graph of the function ##f##. Alternatively, what might a total area of ##0## mean?
 
  • #7
upload_2018-9-18_19-19-49.png

I have drawn the function. An area of zero means that it has to be two areas, one over x-axis, and one under, with the same area?
 

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  • #8
Kolika28 said:
View attachment 230841
I have drawn the function. An area of zero means that it has to be two areas, one over x-axis, and one under, with the same area?

Yes, exactly. Starting from the y-axis in this case. If you changed the lower limit on the integral from ##0##, then you would be calculating the area from a different starting point.
 
  • #9
Kolika28 said:

Homework Statement


You are given the function

f(x)=3x^2-4x-8

a) Find the values of a.
View attachment 230840
Explain the answers using the function.

Homework Equations

The Attempt at a Solution


a^3+2*a^2+8*a=0
a=-2 v a=0 v a=4[/B]

I found the answers, but I don't know how to explain my answers by using the function. What does the zeros of the orginal function has to do with the function I integrated?
Thank you so much for help

If you wrote ##f(x)## correctly, then your integral ##F(a) =a^3 + 2 a^2 + 8a## is incorrect. Your written ##F(a)## has roots ##a = 0, -1 \pm i \sqrt{7}.##
 
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Likes Kolika28
  • #10
Ohh, that gives so much sense. Thank you so much, PeroK. I have one last question, it's not a part of the task, but I'm just wondering. If I were to mark both of areas, how would I do it by looking at the function?
 
  • #11
I'm sorry, Ray Vickson. I made a mistake when writing the equation here on the chat, the answer are correct though. I will fix it now.
 
  • #12
Kolika28 said:
Ohh, that gives so much sense. Thank you so much, PeroK. I have one last question, it's not a part of the task, but I'm just wondering. If I were to mark both of areas, how would I do it by looking at the function?

I'm not sure that you mean. You can see with this function that it's not actually very easy to compare the different areas. In the end, computing the integral is the best way to compare areas, for all but the simplest functions.
 
  • #13
I mean, if I were to express both of the areas with an integral, what would the limits on the integrals be? Would one of them have limits -2 and 0, and the other one would have the limits 0 and 4. I know that's wrong, but how am I supposed to find the limits?
 
  • #14
Kolika28 said:
I mean, if I were to express both of the areas with an integral, what would the limits on the integrals be? Would one of them have limits -2 and 0, and the other one would have the limits 0 and 4. I know that's wrong, but how am I supposed to find the limits?

What you found was that the integral from ##-2## to ##0## is zero; and, the integral from ##0## to ##4## is zero.

You can more or less see this from the graph.
 
  • #15
PeroK said:
What you found was that the integral from ##-2## to ##0## is zero; and, the integral from ##0## to ##4## is zero.

You can more or less see this from the graph.
I understand now. Thank you very much for your help, and not least for your patience. It takes some time for me to understand math, but you really helped me.
 

Related to How Do Zeroes of a Function Relate to Its Integral?

1. What are integrals?

Integrals are mathematical operations that involve finding the area under a curve on a graph. They are used in calculus to solve problems related to finding the total change or accumulation of a quantity over a given interval.

2. Why are integrals important?

Integrals are important because they have a wide range of applications in physics, engineering, economics, and other fields. They can be used to solve problems related to motion, optimization, and probability, among others.

3. How do I solve integrals?

To solve an integral, you can use various methods such as integration by substitution, integration by parts, and the fundamental theorem of calculus. It is important to have a good understanding of the rules and techniques for solving integrals before attempting to solve them.

4. What are the different types of integrals?

The two main types of integrals are definite and indefinite integrals. Definite integrals have specific limits of integration and give a numerical value as a result, while indefinite integrals have no limits and give a general formula as a result.

5. How can I practice and improve my skills in solving integrals?

You can practice solving integrals by working through practice problems and using online resources such as tutorials, videos, and interactive quizzes. It is also helpful to review the different techniques and concepts related to integrals to improve your understanding and skills.

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