8.1 Challenging Integral Applications
Question #5 (Medium): Length of a Trajectory
For functions where evaluating ∫ √ ( ) is challenging, Simpson’s Rule can be used
to approximate the length of the curve for the given interval.
A trajectory of follows after the given equation. Calculate the total distance traveled before hitting the
ground, where is in meters and in seconds.
The interval is not given. So it needs to be determined based on the equation. If it is to count from time
0 until it hits the ground, then ; ; ;
√ √ ; √ since t represents time only positive option works.
Then total distance travelled is essentially equal to the length of the curve. The derivative of the given
function is: . Then ∫ √ ( ) ∫ √ ( )
∫ ( ) . use substitution where , then . Align the