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Vector Algebra

Question
CBSEENMA12032727

Draw a rough sketch of the curves y = sin x and y = cos x as x varies from 0 to straight pi over 2 and find the area of the region enclosed by them and the x-axis.

Solution

Let OB and CA represent the curves y = sin x and y = cos x as x varies from 0 to straight pi over 2. The two curves intersect at D where
                                sinx space equals space cosx space space space space space space space space space space rightwards double arrow space space space tanx space equals space 1 space space space space space space space space space space rightwards double arrow space space straight x space equals space straight pi over 4
Required area = Area OAB + area OCA - area ODA
                         equals space integral subscript 0 superscript straight pi over 2 end superscript sinx space dx space plus space integral subscript 0 superscript straight pi over 2 end superscript cosx space dx minus integral subscript 0 superscript straight pi over 4 end superscript sinx space dx space minus integral subscript straight pi over 4 end subscript superscript straight pi over 2 end superscript cosx space dx
                          equals open square brackets negative cos space straight x space close square brackets subscript 0 superscript straight pi over 2 end superscript plus open square brackets sin space straight x close square brackets subscript 0 superscript straight pi over 2 end superscript minus open square brackets negative cos space straight x close square brackets subscript 0 superscript straight pi over 4 end superscript space minus space open square brackets sin space straight x close square brackets subscript straight pi over 4 end subscript superscript straight pi over 2 end superscript                          equals space open square brackets negative cos space straight pi over 2 plus cos space 0 close square brackets plus open square brackets sin space straight pi over 2 minus sin space 0 close square brackets space minus space open square brackets negative cos straight pi over 4 plus cos space 0 close square brackets minus open square brackets sin straight pi over 2 minus sin straight pi over 4 close square brackets
equals space left parenthesis negative 0 plus 1 right parenthesis plus left parenthesis 1 minus 0 right parenthesis minus open parentheses negative fraction numerator 1 over denominator square root of 2 end fraction plus 1 close parentheses space minus space open parentheses 1 minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses
space equals square root of 2 space square space units.

                        
                           

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