Calculus Question

Question

Your velocity is given by v(t)=1t^2+2 in m/sec, with t in seconds. Estimate the distance, s, traveled between t=0 and t=8. Use the average of the left and right sums with 4 subdivisions. 

Answer: s=

Answer 1

I think you need to show the two sums requested instead of using the midpoint method in the first answer:

Four subdivisions means 8/4 = 2 units in t per division. Make a table t = {0,2,4,6,8} and v = {2,6,18,38,66}. From the concept distance = rate*time, s(left) is the sum of velocities 1 - 4 times the interval of 2 seconds, s(left) = (2+6+18+38)*2 = 128 m; s(right) = (6+18+38+66)*2 = 256 m. The average of these is 192 m, the requested answer. (The exact value from integral calculus is 560/3 = 186 2/3 m; the first tutor got 576/3 via a mistake changing 512 into 528)

Answer 2

v(t)=t^2 +2

s(t) = integral of t^2+2 = (1/3)t^3 + 2t (+ C, but C=0) so

s(t) = (1/3)t^3 + 2t

s(8) = (1/3)(8)^3 + 2(8) = 512/3 + 16 = 528/3 + 48/3 = 576/3 = 192

graphically, s(t) = the area under the parabola s =t^2 + 2

divide it in four divisions, the averages of their x values are 1,3,5 and 7

corresponding y values are 5,13,29 and 53

they sum to 100 x 2 = 200 estimate which is fairly close to the exact 192