Section 5.1 Approximating Areas (and Distance Traveled)
References.
- Calculus, Early Transcendentals by Stewart, Section 5.1.
One of the beauties of mathematics is that often problems that seem to be very different turn out to have very similar mathematical representations and solutions. Two such problems are
- finding the area of a region with curved boundary, and
- finding the distance traveled when we know how the velocity varies with time.
Problem 1: The Area of a Region with Curved Upper Boundary.
We can compute the area shown in the figure where the upper boundary is the curve \(y = f(x)\text{,}\) the lower boundary is the \(x\) axis, the left boundary is the line \(x = a\text{,}\) and the right boundary is the line \(x = b\) as follows:
Divide the interval \([a,b]\) into \(n\) small subintervals each with length
\begin{equation*}
\Delta x = \frac{b-a}{n}
\end{equation*}
and let \(x_0, x_1, x_2, \dots, x_n\) be the end points of the subintervals, so that
\begin{equation*}
x_0 = a, x_i = a+i\Delta x, x_n = b
\end{equation*}
The area \(A_i\) above the \(i\)-th subinterval \(x_{i-1} < x < x_i\) will be approximately the area of a rectangle with width \(\Delta x\) and height \(f(x_i)\text{:}\)
\begin{equation*}
A_i \simeq f(x_i)\Delta x
\end{equation*}
Adding these up we get the total area is given (approximately) by
\begin{equation}
A = \sum_{i=1}^n A_i \simeq R_n := \sum_{i=1}^n f(x_i)\Delta x.\tag{5.1.1}
\end{equation}
This is the so-called right-hand endpoint rule, because we use the value of \(f(x)\) at the right-hand end of each sub-interval \([x_{i-1},x_i]\) as the height of the rectangle over that interval. An alternative is to use the height at the left end of each interval, giving the left-hand endpoint rule
\begin{equation}
A \simeq L_n = \sum_{i=0}^{n-1} f(x_i)\Delta x.\tag{5.1.2}
\end{equation}
Note: never use the value at both the left and right endpoints!
The Exact Area, Using Limits.
If we used more subintervals (larger \(n\) and thus smaller \(\Delta x\)), we could get a better approximation, because the rectangles would fit the true area closer over the shorter intervals. If we can find the limit as \(n \to \infty\) and \(\Delta x \to 0\) of these approximating sums, then we can find the area exactly:
\begin{equation}
A = \lim_{\Delta x \to 0} \sum_{i=1}^n f(x_i)\Delta x, = \lim_{n \to \infty} \sum_{i=1}^n f(x_i)\Delta x\tag{5.1.3}
\end{equation}
It also turns out that the approximations \(L_n\) and \(R_n\) lead to the same limit, so long as \(f(x)\) is continuous. We could also use rectangles with heights given by the function's value at intermediate points in each interval, such as the middle points \(a + h/2\text{,}\) \(a + 3h/2\) ... \(b - h/2\text{.}\) In fact, we use this formula to define area in this situation; without calculus and limits, area has only really been defined for polygons.
Later in this chapter we will learn how to evaluate such limits, at least for some functions \(f\text{.}\) However, the accurate approximations given by the sum formulas with small \(\Delta x\) are often also often useful in practice.
Problem 2: Displacement (net change in position) from Velocity.
If we know a function that gives the velocity of an object at time \(t\text{,}\) that is we know \(v=f(t)\) and we want to find the distance \(s\) that the object travels over a time interval \(a \leq t \leq b\text{,}\) we can proceed as follows: Divide the time interval \([a, b]\) into \(n\) small subintervals each with length
\begin{equation*}
\Delta t = \frac{b-a}{n}
\end{equation*}
and let \(t_0, t_1, t_2, \dots, t_n\) be the end points of the subintervals, so that
\begin{equation*}
t_0 = a, t_i = a+i\Delta t, t_n = b
\end{equation*}
Over the \(i\)-th subinterval \(t_{i-1} < t < t_i\text{,}\) the velocity can be approximated by its value at the start of that interval, \(f(t_{i-1})\text{,}\) so that we can use the familiar formula “distance = rate \(\times\) time” to compute the distance \(s_i\) traveled (approximately) over that short time interval:
\begin{equation}
s_i \simeq f(t_{i-1})\Delta t.\tag{5.1.4}
\end{equation}
Adding these up, the total distance traveled is given (approximately) by
\begin{equation}
s \simeq \sum_{i=0}^{n-1} f(t_i)\Delta t.\tag{5.1.5}
\end{equation}
From Approximations to the Exact Displacement.
If we used more subintervals (larger \(n\) and thus smaller \(\Delta t\)), we could get a better approximation, because the velocity would be closer to being constant over the shorter intervals. If we can find the limit as \(n \to \infty\) and \(\Delta t \to 0\) of these approximating sums, then we can find the distance exactly.
\begin{equation}
s = \lim_{\Delta t \to 0} \sum_{i=1}^n f(t_i)\Delta t, = \lim_{n \to \infty} \sum_{i=1}^n f(t_i)\Delta t.\tag{5.1.6}
\end{equation}
Comparing the Area and Distance Formulas.
Note the similarity between Equation (5.1.3) for area under curve and Equation (5.1.6) for computing distance from a velocity function: this allows the same methods to be used for both the area and the distance problems, for both approximation and exact evaluation.
Indeed a great variety of the mathematical and scientific problems can be solved in terms of the same sort of “limit of a sum” formula, which makes evaluation of this quantity of great importance.
This is the topic for the rest of this course, and a major topic in Calculus II.
Exercises Exercises
Study Calculus Volume 1, Section 5.1 2 ; in particular, if you are unfamiliar with the \(\Sigma\) notation for sums, the first part of that section should help. Study Exercises 15, 19, 23, 27, 29, and 43.
openstax.org/books/calculus-volume-1/pages/5-1-approximating-areasopenstax.org/books/calculus-volume-1/pages/5-1-approximating-areas
