solution:
s( t ) = ( 2 )sin( πt ) + ( 2 )cos(πt )
v( t ) = s'( t ) = ( 2π )cos( πt) - ( 2π )sin( πt )
vavg = 1 / ( b - a ) Integral a to b [ v( t ) ] dt
( a )
vavg
= 1 / ( 2 - 1 ) Integral 1 to 2 [ ( 2π)cos( πt ) - ( 2π )sin( πt ) ] dt
= [ ( 2)sin( πt ) + ( 2 )cos( πt ) ] 1to 2
= [ ( 2)sin( 2π ) + ( 2 )cos( 2π ) ] - [ ( 2 )sin( π ) + ( 2 )cos( π) ]
= 4 cm / s
( b )
vavg
= 1 / ( 1.1 - 1 ) Integral 1 to 1.1 [ ( 2π)cos( πt ) - ( 2π )sin( πt ) ] dt
= 10 [ ( 2 )sin( πt ) + ( 2 )cos( πt ) ] 1 to 1.1
= 10 [ [ ( 2 )sin( 1.1π ) + ( 2)cos( 1.1π ) ] - [ ( 2 )sin( π) + ( 2 )cos( π ) ] ]
-5.20 cm /s
( c )
vavg
= 1 / ( 1.01 - 1 ) Integral 1 to 1.01 [ ( 2π)cos( πt ) - ( 2π )sin( πt ) ] dt
= 100 [ ( 2 )sin( πt ) + ( 2 )cos( πt ) ] 1 to 1.01
= 100 [ [ ( 2 )sin( 1.01π ) + (2 )cos( 1.01π ) ] - [ ( 2 )sin( π) + ( 2 )cos( π ) ] ]
-6.18 cm /s
( d )
vavg
= 1 / (1.001 - 1 ) Integral 1 to 1.001 [ ( 2π )cos( πt ) - ( 2π )sin(πt ) ] dt
= 1000 [ ( 2 )sin( πt ) + ( 2 )cos( πt ) ] 1 to 1.001
= 1000 [ [ ( 2 )sin( 1.001π ) + (2 )cos( 1.001π ) ] - [ ( 2 )sin(π ) + ( 2 )cos( π ) ] ]
-6.27 cm /s
