Word problem with integral

Question

A train travels along a track and its speed (in miles per hour) is given by s(t)=65t for the first half hour of travel. Its speed is constant and equal to s(t)=65/2 after the first half hour. Here time t is measured in hours. How far (in miles) does the train travel in the first hour of travel?

Answer 1

x(t) = ∫65tdt(from 0 to .5) + ∫65/2dt(from .5 to 1)

 

First part of equation:

∫65t from 0 to .5 --> (.5 being a half hour) 

65t²/2 from 0 to .5 = 65(.5)²/2 - 0 = 8.125 miles

 

Second part of equation:

∫65/2 from .5 to 1 = 65t/2 = 65(1)/2 - 65(.5)/2 = 32.5 - 16.25 = 16.25 miles

 

total distance is equal to 8.125 + 16.25 miles = 24.375 miles