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Given the velocity function f(t)=e^(-t^(2)//2),a=0,b=2,n=30Here, f(t) is an exponential function and hence, velocity is always positive. So, distance will be equal to the displacement as calculated below.width of rectangle (Delta t)=(2-0)/(30)=0.067also., to =0 and t_(30)=2.Now, as we know,{:[" Left hand sum "=sum_(i=0)^(n-1)f(t_(i))Delta t.],[=(f(t_(0))+f(t_(1))+f(t_(2))+cdots+f(t_(27))+f(t_(28))+^(N)*:}],[{:f(t_(29)))Delta t]:}Also,{:[" Right - hand sum "=sum_(i=1)^(n-1)f(t_(i))Delta t.],[=(f(t_(1))+f(t_(2))+f(t_(3))+cdots+f(t_(2))+f(t_(29))+f(t_(30)))Delta t]:}As we can observe from the terms of left hand and right hand sums written above, following terms are common in both (f(t_(1))+f(t_(2))+cdots+f(t_(2))+f(t_(2g))) st.Now, one of the left and right-hand sum will be lower estimate and other one will be upper es ... See the full answer