 Let's do a quick review of the main ideas of section 4.1 on determining distance traveled from velocity. We started our study of calculus with the following problem. Given a moving object and a function that tells us its position at a given time, how fast is it moving at a particular single moment? Here we're asking a sort of reverse question. Given a moving object and its velocity function, how far does it travel from one time value to another? So, whereas with derivatives, we're given a position function and find velocity. Here, in the new question, we're going to be starting with velocity and we'd like to find something having to do with a change in position. We're going to spend the rest of chapter four refining our answers to this question. Just like we spent the majority of chapters one and two, we're finding our answer to the first question. But here's the basic idea we will develop at section 4.1. If you look at an object's velocity graph and you want to find the distance traveled from t equals a to t equals b, we can approximate that distance traveled by drawing a single rectangle whose base spans from t equals a to t equals b on the horizontal axis. As I have here drawn this rectangle going from t equals 1 to t equals 3. And whose height equals the function value of the left hand end point of that interval, which in this case is t equals 1. The area of that rectangle is the base of the rectangle times the height of the rectangle as we know. But notice that the height is actually a velocity value, and the width is actually a length of time. So we can interpret the area of this rectangle as velocity times time, which gives us distance, according to that old rule that says distance equals rate times time. The area of this rectangle, therefore, approximates the distance traveled on this interval. It's an approximation and not something exact because the velocity is not the same throughout the whole time interval. We can improve this estimate, though, by sampling the object's velocity more frequently in the time interval. Instead of using one rectangle, we split the interval up into several subintervals, create several rectangles, find the height of each using the velocity graph, and then find the area of each, and then add all those areas up. Each quote unquote area here is really a distance traveled. So this sum is an estimate to the total distance traveled. And as you can see, with more rectangles, we get a better estimate than with fewer rectangles. So geometry is one way to determine distance traveled from velocity. Another way is to notice the connection to derivatives. In this situation that's new, we're given velocity, which is the derivative of position. And we can find the distance traveled by working backwards to find the original position function. This process of starting with a derivative and working backwards to find the original function is called anti-differentiation. And the function we get is called an anti-derivative of the velocity function. More generally, f is said to be an anti-derivative of g if f prime equals g. That is, we start with g and we work our way back to find f. If we start with a velocity function and then anti-differentiated, then the distance traveled is just going to be the difference between the starting and ending positions. That is, the anti-derivative evaluated at the ending time minus the anti-derivative evaluated at the starting time. So anti-differentiation is an awesome tool for finding distance traveled. But as we'll see, it's also fraught with computational issues. And so we want to keep the more geometric approach very close at hand.