Calculus problem involving distance traveled!?

Question

For time t, in hours, 0 t 1, a bug is crawling at a velocity, v, in meters/hour given by the following equation:
V= 1/(0.83 + t)
Use delta t = 0.2 to estimate the distance that the bug crawls during this hour. Find an overestimate and an underestimate. Then average the two to get a new estimate.
meters (overestimate) =
meters (underestimate) =
meters (new estimate) =
Please explain how to do the problem and how you arrived at the answers given.

Answer 1

Note that dv/dt = -1/(0.83 + t)² which is negative for all t. That means that the rate of change of the velocity is negative; that is, the velocity is decreasing with time. The bug is slowing down. Knowing this will help us choose v-values to produce an overestimate and an underestimate of the true distance traveled.

The true distance that the bug travels is

integral from 0 to 1 of v(t) dt = integral from 0 to 1 of 1/(0.83 + t) dt.

You're supposed to approximate this with Riemann sums using Δt = 0.2. Because v is decreasing, using v-values on the left side of the intervals will produce an overestimate, and using v-values on the right side of the intervals will produce an underestimate.

The overestimate using left-side values is

[ v(0) + v(.2) + v(.4) + v(.6) + v(.8) ] Δt

and the underestimate using right-side values is

[ v(.2) + v(.4) + v(.6) + v(.8) + v(1.0) ] Δt

I get about 0.86 meter for the overestimate and about 0.73 meter for the underestimate. The true value is about 0.7906 meter.

Answer 2

So as t goes from 0 to 1, the bug slows down because the bottom gets bigger. The requirement to have delta t = 0.2 means there will be 5 intervals since 1 hour divided by 0.2 is 5.

You get the overestimate by starting at 0 for t, then going forward: 0, 0.2, 0.4, 0.6 and 0.8 for the 5 times. You get the underestimate by starting at 1 and going backwards, 1, 0.8, 0.6, 0.4 and 0.2.

Plug these values in for t, into the distance formula d = vt, add them up to get over and under, then add these and divide by 2 to get the new estimate.

I'll start under for you: 1/(0.83+1)*1 + 1(0.83 + 0.8)*0.8 + 1/(0.83 + 0.6)*0.6 + 1/(0.83 + 0.4)*0.4 + 1/(0.83 + 0.2) * 0.2=

No calculator here so you're on your own [* means multiply]