we have to find total length travelled as a function of t.... let the distance travelled in a small time dt be dl then the x and y coordinates change by dx and dy respectively as shown(both are non constant functions of t) therefore by pythagoras theorem the length dl =sqrt(dx2+dy2) but x=t+sint y=cost...differentiating....dx=(1+cost)dt dy=-sintdt...therefore dl=sqrt(dx2+dy2)=sqrt(((1+cost)dt)2+(sintdt)2)=sqrt(cos2t+sin2t+1+2cost)dt=sqrt(2+2cost)dt...therefore dl=sqrt(2+2cost)dt but the function changes sign ...therefore we have to find areas separately and add the moduli..(otherwise areas will get subtracted instead of added)... now integrating... =dl= 2sqrt(1+cost)dt= 2(cos(t/2))dt=4sin(t/2)+C...but C=0..now between t=0 to t= area =4 and between t= and t=2 the modulus of area is 4 therefore total area is 8.....now its right...
Moderation Team
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