Section 5.2 The Definite Integral
The ``limit of sums'' formula seen in Section 5.1 for computing both distance traveled and area under a curve is also useful in many other cases, and the main goal of this chapter is to learn more about how to do this calculation in practice, without having to actually evaluate the sums or limits, but instead mostly using anti-derivatives.
In this section we make a careful statement of the quantity to be calculated, introduce some variants on the Riemann sum approximation of the area under a curve to make calculator approximations more accurate and efficient, and learn some properties akin to those for limits, derivatives and anti-derivatives: rules for sums, differences, constant multiples, etc.
A key calculus strategy: first approximate, then find a limit.
There are many other problems that can be calculated by the above process of
approximating a quantity by a sum of function values times a small interval width \(\Delta x\text{,}\) and then
finding the exact quantity as the limit as the number of function values used goes to \(\infty\) and \(\Delta x\) goes to 0.
Thus we need a name and notation for it:
Definition 5.2.1. Definite Integral, right-hand rule version.
For \(f(x)\) is a continuous function on the interval \(a \leq x \leq b\text{,}\) the definite integral of \(f(x)\) over the interval \([a,b]\text{,}\) denoted \(\displaystyle\int_a^b f(x) \, dx\text{,}\) is the numerical value given by the limit
\begin{equation*}
\int_a^b f(x) \, dx
= \lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x,
= \lim_{\Delta x \to 0} \sum_{i=1}^n f(x_i) \Delta x,
\end{equation*}
where \(\Delta x = \frac{b-a}{n}\) and \(x_i = a + i \Delta x\text{,}\) so \(x_0=a\) and \(x_n=b\text{.}\)
Note that the notation
modifies the ``Greek S'' \(\Sigma\) to become the ``elongated S'' \(\int\text{,}\) and
changes the ``Greek D'' \(\Delta\) in \(\Delta x\) to the ``small d'' in \(dx\text{,}\) to indicate that the limit was taken as \(n \to \infty\) (i.e. \(\Delta x \to 0\)).
Other Choices for the Rectangle Heights and Widths.
The sums of areas of rectangles used above to approximate the area under the curve is called a Riemann Sum, but the choice of using intervals of equal width with the height of each rectangle being the height of the curve at the right endpoint of each interval is not the only possibility: it was used partly because it makes the notation easiest. The intervals can instead vary in width, and the heights can instead be computed at other points \(x_i^{*}\) in each interval, like the left endpoints or the midpoints, or a different choice in each interval. Of these options, using the mid-point of each interval is intuitively the best choice, and this in fact can be proven to be the most accurate in some sense, to be seen in Calculus 2.
The most general form of the approximation for area under the curve allows for the interval \([a,b]\) to be divided into possibly unequal intervals by \(x\) values \(a=x_0 < x_1 \cdots < x_n = b\text{,}\) with widths
\begin{equation*}
\Delta x_1 = x_1-x_0, \dots , \Delta x_i = x_i-x_{i-1}, \dots , \Delta x_n = x_n-x_{n-1},
\end{equation*}
taking any point \(x_i^*\) within each sub-interval \([x_{i-1},x_i]\) to get the height of a rectangle on that sub-interval. Then the approximate area under the curve is the general Riemann sum Approximation
\begin{equation*}
\sum_{i=1}^n f(x_i^{*}) \Delta x_i = f(x_1^*) \Delta x_1 + f(x_2^*)\Delta x_2 \cdots f(x_n^*)\Delta x_n.
\end{equation*}
It can be shown that even with varying intervals widths and choices of where in each interval to compute the rectangle height, the approximations all get close to the same value when the widths of all the subintervals are very small (no rectangle width \(\Delta x_i\) bigger than some maximum width \(\Delta x\)), so long as \(f(x)\) is continuous on \([a,b]\text{.}\) The proof is omitted here; it is seen is advanced calculus courses.
This gives the most general definition:
Definition 5.2.2. Definite Integral, with all Riemann Sum Approximations.
If \(f(x)\) is a continuous function on the interval \(a \leq x \leq b\text{,}\) with the \(x_i\text{,}\) \(\Delta x_i\) and \(x_i^*\) as above and \(\Delta x_i \leq \Delta x\text{,}\) then
\begin{equation*}
\int_a^b f(x) \, dx = \lim_{\Delta x \to 0} \sum_{i=1}^n f(x_i^*) \Delta x_i
\end{equation*}
The Mid-point Rule.
Of these approximations, using the mid-point of each interval is intuitively the best choice. % and this in fact can be proven to be the most accurate in some sense. %, to be seen in Calculus 2. It is still simplest to use \(n\) intervals of equal width \(h=\Delta x = (b-a)/n\text{,}\) which gives the \(n\)-point midpoint rule approximation
\begin{equation*}
\int_a^b f(x) \, dx
\approx M_n
= h \sum_{i=1}^n f\left(\frac{x_{i-1}+x_i}{2}\right)
= h \sum_{i=1}^n f(a+(i-1/2)h),
\end{equation*}
where \(x_i = a+i h\) (\(x_{0}=a, x_1=a+h\text{,}\) etc.)
This sum can be evaluated on calculators with something like
sum(seq(f(a+(i-0.5)*h),i,1,n))*h
or the slightly quirky but easier to type version
sum(seq(f(x),x,a+h/2,b,h))*h
This uses \(x\) values \(a+h/2\text{,}\) \(a+3h/2\) and so on, continuing so long as the value is less than \(b\text{.}\) Actually the last value used is \(b-h/2\text{,}\) but using the upper limit of \(b\) is safer; if you use \(b-h/2\text{,}\) a slight rounding error can cause that last \(x\) value to be omitted!
Properties of the Definite Integral.
Thinking of definite integrals as areas under curves or displacements given by velocities, the following facts are intuitive. We will soon see a simple way to verify them, using anti-derivatives.
\(\displaystyle \int_a^b c \; dx = c(b-a)\) where \(c\) is any constant.
\(\displaystyle \int_a^b c f(x)\; dx = c \int_a^b f(x) \; dx\) where \(c\) is any constant.
\(\displaystyle \displaystyle \int_a^b f(x) + g(x) \; dx = \int_a^b f(x) \; dx + \int_a^b g(x) \; dx\)
\(\displaystyle \displaystyle \int_a^b f(x) - g(x) \; dx = \int_a^b f(x) \; dx - \int_a^b g(x) \; dx\)
\(\displaystyle \displaystyle \int_a^b f(x)\; dx + \int_b^c f(x)\; dx = \int_a^c f(x)\; dx\)
\(\displaystyle \displaystyle \int_b^a f(x)\; dx = - \int_a^b f(x) \; dx\)
\(\displaystyle \displaystyle \int_a^a f(x)\; dx = 0\)
Comparison Properties of the Definite Integral.
If
\(f(x) \geq 0\) for
\(a \leq x \leq b\text{,}\) then
\begin{equation*}
\int_a^b f(x)\; dx \geq 0.
\end{equation*}
If
\(f(x) \geq g(x)\) for
\(a \leq x \leq b\text{,}\) then
\begin{equation*}
\int_a^b f(x)\; dx \geq \int_a^b g(x)\; dx.
\end{equation*}
If
\(m \leq f(x) \leq M\) for
\(a \leq x \leq b\text{,}\) then
\begin{equation*}
m(b-a) \leq \int_a^b f(x)\; dx \leq M(b-a).
\end{equation*}
The last is a cousin of the Mean Value Theorem.
Exercises Exercises
Study
Calculus Volume 1, Section 5.2 2 ; in particular Examples 8 to 13, Checkpoints 8 to 12, and Exercises 61, 65, 73, 75, 79, 81, 89, 91, 93, 99, 101 and 107.
openstax.org/books/calculus-volume-1/pages/5-2-the-definite-integral
openstax.org/books/calculus-volume-1/pages/5-2-the-definite-integral
