Let I=∫ _ a ^ a x√( a^ 2 x^ 2 ) dx And f( x ) =x√( a^ 2 x^ 2 ) Then f( x ) =( x ) √( a^ 2 ( x ) ^ 2 ) = x√( a^ 2 x^ 2 ) = f( x ) ∴ ...

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Integration of Piecewise Continuous Functions

∫-aax√a2-x2dx...

Question

$\int_{- a}^{a} x \sqrt{a^{2} - x^{2}} d x = ? ?$

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Solution

Let $I = \int_{- a}^{a} x \sqrt{a^{2} - x^{2}} d x$
And $f (x) = x \sqrt{a^{2} - x^{2}}$
Then $f (- x) = (- x) \sqrt{a^{2} - {(- x)}^{2}} = - x \sqrt{a^{2} - x^{2}} = - f (x)$
$∴ I = 0$

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