Question

In Exercises \(29-32,\) express the desired quantity as a definite integral and evaluate the integral using Theorem \(2 .\) Find the distance traveled by a train moving at 87 mph from \(8 : 00\) A.M. to \(11 : 00\) A.M.

Short answer

The distance traveled by the train from \(8:00\) AM to \(11:00\) AM is \(261\) miles.

Step-by-step solution

Expressing Quantity as Definite Integral

We know that distance defined by a constant speed is the area under the speed-time graph. For a constant speed, the graph would be a straight horizontal line for the given speed that extends for the given time period. Therefore, the distance can be represented as the definite integral of speed from the start to end times, that is \(8:00\) AM to \(11:00\) AM or, changing it to hours gives us, from 0 to 3 (assuming 8:00 AM as t=0). So, the integral would be \(\int_{0}^{3} 87 dt\).

Evaluating the Definite Integral

As the integrand, \(87\) is merely a constant, the integral simply becomes \(87 \times t |_{0}^{3}\). Evaluating it at the limiting points:

Calculation

Now perform the calculation. The result is \(87 \times 3 - 87 \times 0 = 261\) miles.