Absolute value definite integral

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Homework Help Overview

The discussion revolves around evaluating the definite integral of the absolute value function ∫[|2x-1|] over the interval [1,-1]. Participants are exploring how to properly split the integral based on the behavior of the absolute value function within the specified limits.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the need to split the integral into two parts based on the sign of the expression inside the absolute value. They raise questions about determining the correct intervals for each piece of the function, specifically where to apply 2x-1 and where to apply 1-2x.

Discussion Status

Some participants have identified the critical point at x = 1/2, which helps in determining the intervals for the piecewise function. There is ongoing clarification about how to apply these intervals in the context of the integral evaluation.

Contextual Notes

Participants are working within the constraints of evaluating a definite integral involving absolute values, and there is a focus on understanding the conditions under which each piece of the function is valid.

Mdhiggenz
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Homework Statement



∫[|2x-1|] evaluated on the interval [1,-1]

Homework Equations





The Attempt at a Solution



I know we must split it into two equations ∫ from [a,c] + ∫ from [c,b]

and in absolute value one is negative and one is positive, so it will be

2x-1, and 1-2x my question is how do we know which interval to place each one in.

Do we place 1-2x in the interval from [c,b] or 2x-1, and vice versa,

thank you
 
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Mdhiggenz said:

Homework Statement



∫[|2x-1|] evaluated on the interval [1,-1]

Homework Equations





The Attempt at a Solution



I know we must split it into two equations ∫ from [a,c] + ∫ from [c,b]

and in absolute value one is negative and one is positive, so it will be

2x-1, and 1-2x my question is how do we know which interval to place each one in.

Do we place 1-2x in the interval from [c,b] or 2x-1, and vice versa,

thank you
Where is 2x-1 ≥ 0 ?

Where is 1-2x ≥ 0 ?
 
for 2x-1 when x is greater than or = to 1/2

and for 1-2x when x is less than or = to 1/2

?
 
Mdhiggenz said:
for 2x-1 when x is greater than or = to 1/2

and for 1-2x when x is less than or = to 1/2

?
That should allow you to evaluate \displaystyle \int_{-1}^1\ |2x-1|\,dx\,.
 
Ohh so what you are saying is since 1-2x is less than 1/2 it goes in the interval from [0,1/2]?
 

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