Definite Integral of an Absolute Value Function

Limits, differentiation, related rates, integration, trig integrals, etc.
Hilariousity
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Definite Integral of an Absolute Value Function

Postby Hilariousity » Sun Apr 17, 2011 12:44 am

Hi Everybody,
I'm having problems finding the area between the x-axis and the function |2x-3| on the domain (-1,2). So what I have been trying to do is integrate |2x-3| to get 2x^2/2-3x. Then plug in 2-(-1)=3 into the equation to obtain 0. This is incorrect. The answers that are written in the back of the book tell me that the answer should be 13/2. What am I doing wrong?

nona.m.nona
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Re: Definite Integral of an Absolute Value Function

Postby nona.m.nona » Sun Apr 17, 2011 12:57 am

Hilariousity wrote:I'm having problems finding the area between the x-axis and the function |2x-3| on the domain (-1,2). So what I have been trying to do is integrate |2x-3| to get 2x^2/2-3x.

Look at the graph of f(x) = |2x - 3|. It is not the same as the graph of g(x) = 2x - 3, so the integral, naturally, should not be the same.

Instead, split the absolute-valued function into two standard linear functions, one on the left-hand portion of the interval, where the original graph is decreasing, and the other on the right-hand portion of the interval, where the original graph is increasing.

Do the two integrations over the two intervals. Sum to find the value of the integral of the original function.


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