Compute, please, definite integral which have a absolute value function

ushwaui 2021-11-19 Answered
Compute, please, definite integral which have a absolute value function like these below:
\(\displaystyle{\int_{{-{2}}}^{{{3}}}}{\left|{x}\right|}{\left.{d}{x}\right.}\)
\(\displaystyle{\int_{{-{2}}}^{{{3}}}}{\left|{x}-{1}\right|}{\left.{d}{x}\right.}\)
\(\displaystyle{\int_{{-{2}\pi}}^{{{2}\pi}}}{\left|{\sin{{x}}}\right|}{\left.{d}{x}\right.}\)

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Expert Answer

Elizabeth Witte
Answered 2021-11-20 Author has 7440 answers

You should consider the fact that |x| is one linear function for \(\displaystyle{x}\in{\left(-\infty;{0}\right]}\) and a linear function for \(x\in[0;\infty)\) to break the interval of integration into subintervals that are easier.
\(\displaystyle{\int_{{-{2}}}^{{{3}}}}{\left|{x}\right|}{\left.{d}{x}\right.}={\int_{{-{2}}}^{{{0}}}}-{x}{\left.{d}{x}\right.}+{\int_{{{0}}}^{{{3}}}}{x}{\left.{d}{x}\right.}\)

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Crom1970
Answered 2021-11-21 Author has 8385 answers
Firstly, you can split each integral into multiple, smaller integrals for your first integral, such as:
\(\displaystyle{\int_{{-{2}}}^{{{0}}}}-{x}{\left.{d}{x}\right.}\) and \(\displaystyle{\int_{{{0}}}^{{{3}}}}{x}{\left.{d}{x}\right.}\)
You can also look at the graphs and calculate the area underneath them using simple geometry.
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