The path of a drone is given in meters by and for

MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

The path of a drone is given in meters by

x=2t^{2}

and

y=3t-\tfrac{1}{3}t^{3}

for

0\le t\le2

, where

t

is measured in seconds. Because the antiderivative of the integrand needed for the exact length is not elementary, you will estimate the distance traveled. (a) Write, but do not evaluate, the definite integral whose value equals the total distance traveled by the drone from

t=0

t=2

. (b) The table below lists the values of

t

at four equally spaced points and the corresponding numerical values of

\sqrt{\bigl(\frac{dx}{dt}\bigr)^2+\bigl(\frac{dy}{dt}\bigr)^2}

(all in meters per second): |

t

|\mathbf{v}(t)|

| | --- | ----------------- | | 0.0 | 3.000 | | 0.5 | 3.216 | | 1.0 | 3.873 | | 1.5 | 5.044 | | 2.0 | 6.816 | Use Simpson’s Rule with

n=4

subintervals to approximate the distance traveled. Show all calculations and include units. (c) Briefly discuss whether your estimate in part (b) is likely an overestimate or an underestimate, citing the concavity of the integrand on

0\le t\le2

in your reasoning.

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