Integrate[Abs[t], {t, -a, a}] == a Sqrt[Im[a]^2 + Re[a]^2]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mi> a </mi> </msubsup> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> t </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </lowlimit> <uplimit> <ci> a </ci> </uplimit> <apply> <abs /> <ci> t </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <imaginary /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29