- #1
Prasun-rick
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Member warned that homework questions must be posted in the HW & CW sections
∫|x^3-x|dx with limits from -1 to 2.Can anyone kindly show the step wise solution or suggest how to proceed .Thanks in advance
How do you solve the definite integral ∫|x^3-x|dx with limits from -1 to 2?
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In summary, the conversation is about finding the integral of |x^3-x| with limits from -1 to 2. The problem lies in understanding how to approach the modulus sign. The suggested step is to make a drawing to investigate the function and its symmetry. The zero points are -1, 0, and 1. The function is positive between -1 and 0 and there is antisymmetry between 0 and 1. The solution involves splitting up the function and creating separate integrals for each section.
- #2
S.G. Janssens
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Can you tell us first what makes it troubling for you?
- #3
BvU
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First step: make a drawing !
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Prasun-rick
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The modulus sign! I In which limit I have to take positive and negative in the modulus I could not understand!Krylov said:
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Prasun-rick
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Well I am not able to draw precisely and that is where the problem lies !BvU said:
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BvU
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Doesn't have to be precise. Simply investigate the function a little.
Where it comes from on the left, where it goes through zero and where it goes to on the right.
Perhaps there is even a little symmetry to explore and make good use of ...
Where it comes from on the left, where it goes through zero and where it goes to on the right.
Perhaps there is even a little symmetry to explore and make good use of ...
- #7
Prasun-rick
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well kindly please tell me the terminal points ! I found it as -1,0 and 1 am I correct??BvU said:Doesn't have to be precise. Simply investigate the function a little.
Where it comes from on the left, where it goes through zero and where it goes to on the right.
Perhaps there is even a little symmetry to explore and make good use of ...
- #8
BvU
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I don't know what terminal points are but the zero points of ##\ \ x^3-x = x(x+1)(x-1) \ \ ## are indeed -1, 0, 1 .
Between -1 and 0 two out of three factors are < 0 and the other is > 0, so the expression is positive.
Between 0 and 1 ...
There is antisymmetry : ##\ \ (-x)^3 -(-x) = -(x^3 - x) ##, so ...
but the zero points of ##\ \ x^3-x = x(x+1)(x-1) \ \ ## are indeed -1, 0, 1 .Between -1 and 0 two out of three factors are < 0 and the other is > 0, so the expression is positive.
Between 0 and 1 ...
There is antisymmetry : ##\ \ (-x)^3 -(-x) = -(x^3 - x) ##, so ...
- #9
Prasun-rick
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Thanks I finally got that !Thank you very much !BvU said:I don't know what terminal points are
Between -1 and 0 two out of three factors are < 0 and the other is > 0, so the expression is positive.
Between 0 and 1 ...
There is antisymmetry : ##\ \ (-x)^3 -(-x) = -(x^3 - x) ##, so ...
- #10
chiro
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Hey Prasun-rick.
You will have to split up the function within the absolute value function so that it is positive and negative and create separate integrals for each section.
Just remember that you define |x| = x if x > 0 and -x if x < 0 (and 0 if x = 0).
You will have to split up the function within the absolute value function so that it is positive and negative and create separate integrals for each section.
Just remember that you define |x| = x if x > 0 and -x if x < 0 (and 0 if x = 0).
What is a definite integral?
A definite integral is a mathematical concept used to calculate the area under a curve on a graph. It is represented by the symbol ∫ and has a lower and upper limit, indicating the range of values to be integrated.
Why is a troubling definite integral difficult to solve?
A troubling definite integral can be difficult to solve because it may involve complex mathematical functions, require advanced techniques, or have no known analytical solution. In some cases, numerical methods or approximations may be used to solve it.
How is a troubling definite integral different from a regular definite integral?
A troubling definite integral is different from a regular definite integral because it is usually more challenging to solve, either due to its complexity or lack of known solutions. It may also require special techniques or involve unconventional functions.
What are some tips for solving a troubling definite integral?
Some tips for solving a troubling definite integral include breaking it down into smaller, more manageable integrals, using substitution or integration by parts, and utilizing special properties or symmetry of the integrand. It can also be helpful to review common integration techniques and practice solving similar problems.
Why is it important to solve troubling definite integrals?
Solving troubling definite integrals is important in many fields of science, such as physics, engineering, and economics, as it allows for the calculation of important quantities such as area, volume, and displacement. It also helps to understand the behavior of complex functions and can lead to new mathematical insights and discoveries.
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