Question

The area bounded by the curve y = cos x and y = sin x between the ordinates x = 0 and x = 3π/2 is

$left parenthesis 4 minus square root of 2 minus 2 right parenthesis space sq space units$
$left parenthesis 4 minus square root of 2 plus 2 right parenthesis space sq space units$
$left parenthesis 4 minus square root of 2 minus 1 right parenthesis space sq space units$
$left parenthesis 4 minus square root of 2 plus 1 right parenthesis space sq space units$

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Solution

left parenthesis 4 minus square root of 2 minus 2 right parenthesis space sq space units

integral subscript 0 superscript 3 straight pi divided by 2 end superscript vertical line sin space straight x space minus space cos space straight x vertical line space dx space space... space left parenthesis straight i right parenthesis space equals space integral subscript 0 superscript straight pi divided by 4 end superscript space left parenthesis cos space straight x space minus space sin space straight x space right parenthesis space dx space plus space integral subscript straight pi divided by 4 end subscript superscript 5 straight pi divided by 4 end superscript space left parenthesis sin space straight x space minus space cos space straight x space right parenthesis space dx plus integral subscript 5 straight pi divided by 4 end subscript superscript 3 straight pi divided by 2 end superscript space left parenthesis cos space straight x space minus space sin space straight x right parenthesis space dx left square bracket space sin space straight x space plus space cos space straight x right square bracket subscript 0 superscript straight pi divided by 4 end superscript space plus space left square bracket negative cos space straight x space minus space sin space straight x space right square bracket space subscript straight pi divided by 4 end subscript superscript 5 straight pi divided by 4 end superscript space plus space left square bracket space sin space space straight x space space plus space cos space straight x space right square bracket subscript 5 straight pi divided by 4 end subscript superscript 3 straight pi divided by 2 end superscript square root of 2 space minus 1 space plus space square root of 2 space end root space plus space square root of 2 space plus space square root of 2 space minus space 1 space space equals space 4 square root of 2 space minus space 2 space

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