#"we can solve either using the average value of a function,"#
#"or using definition of the average speed"#
#"solution using definition of average speed:"#
#"................................................................#
#v(t)=int a(t)*d t#
#v(t)=int (t^2-t)*d t#
#v(t)=1/3*t^3-1/2*t^2+C#
#"So " t=0" ; "v=3 m/s" ; "C=3#
#v(t)=t^3/3-t^2/2+3" *"#
#Delta s=int _0^4 v(t)*d t#
#Delta s=int_0^4 (t^3/3-t^2/2+3)d t#
#Delta s=|1/3*1/4*t^4-1/2*1/3*t^3+3*t|_0^4#
#Delta s=(1/12*4^4-1/6*4^3+3*4)-(1/12*0-1/6*0+3*0)#
#Delta s=(256/12-64/6+12)-0#
#Delta s=(256-128+144)/12=272/12" m , total distance covered(t=0;t=4)"#
#v_a=("total distance covered")/("given time interval")" definition of average velocity"#
#v_a=(272/12)/(4-0)=272/(4*12)=272/48=5.67" "m/s#
#"solution using the average value theorem:"#
#"................................................................#
#"using speed v(t) * function above"#
#v_a=1/(4-0)int_0^4 v(t)*d t #
#v_a=1/4int_0^4 (t^3/3-t^2/2+3)*d t#
#v_a=1/4[|1/3*1/4*t^4-1/2*1/3*t^3-3*t|_0^4]#
#v_a=1/4[(1/12*4^4-1/6*4^3+3*4)-(0)]#
#v_a=1/4[272/12]#
#v_a=5.67" "m/s#
