calculus question

Question

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.

___ ft

__ft

t (s)	0	0.5	1.0	1.5	2.0	2.5	3.0
v (ft/s)	0	5.7	9.2	14.1	17.5	19.4	20.2

Answer 1

Hi Sonam! Because the velocity is increasing, when using LEFT Riemann sums, the area (the area under a velocity curve is displacement or distance) is an underestimate, and when using RIGHT Riemann sums, the area is an overestimate.

For this problem, doing LEFT Riemann sums starts from t = 0 to t = 2.5 seconds with each Δt = 0.5 seconds.

Distance = 0.5 * v (0) + 0.5 * v(0.5) + ......... 0.5 * v(2) + 0.5 * v(2.5) >>>>>>>>>>>> (LOWER LIMIT)

Same thing can be done with RIGHT Riemann sums from t = 0.5 seconds to t = 3 seconds, with Δt = 0.5

Distance = 0.5 * v (0.5) + 0.5 * v(1) + ......... 0.5 * v(2.5) + 0.5 * v(3) >>>>>>>>>>>> (UPPER LIMIT)

Hope this helped. Let me know in the comments if you have additional questions!