Question

Show that:
$integral subscript 0 superscript 2 straight x square root of 2 minus straight x end root space equals fraction numerator 16 square root of 2 over denominator 15 end fraction$

Solution

Let I = $integral subscript 0 superscript 2 straight x square root of 2 minus straight x end root dx$
$equals space integral subscript 0 superscript 2 left parenthesis 2 minus straight x right parenthesis space square root of 2 minus left parenthesis 2 minus straight x right parenthesis space dx end root$ $open square brackets because space space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis space dx close square brackets$
$equals space integral subscript 0 superscript 2 left parenthesis 2 minus straight x right parenthesis space square root of straight x dx space equals integral subscript 0 superscript 2 open parentheses 2 straight x to the power of 1 half end exponent minus straight x to the power of 3 over 2 end exponent close parentheses dx space equals space open square brackets 2 space fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction minus fraction numerator straight x to the power of begin display style 5 over 2 end style end exponent over denominator begin display style 5 over 2 end style end fraction close square brackets subscript 0 superscript 2 space equals space open square brackets 4 over 3 straight x to the power of 3 over 2 end exponent minus 2 over 5 straight x to the power of 5 over 2 end exponent close square brackets subscript 0 superscript 2 equals space open parentheses 4 over 3 2 to the power of 3 over 2 end exponent minus 2 over 5 2 to the power of 5 over 2 end exponent close parentheses minus left parenthesis 0 minus 0 right parenthesis space equals 4 over 3 cross times 2 square root of 2 minus 2 over 5 cross times 4 square root of 2 space equals 8 over 3 square root of 2 minus 8 over 5 square root of 2 equals space fraction numerator 40 square root of 2 minus 24 square root of 2 over denominator 15 end fraction space equals space fraction numerator 16 square root of 2 over denominator 15 end fraction$

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Integrals

Show that:
$integral subscript 0 superscript 2 straight x square root of 2 minus straight x end root space equals fraction numerator 16 square root of 2 over denominator 15 end fraction$

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