Nuances of the Integral
Let's discuss a few nuances of the definition of the integral from the previous activity.
Nuance 1: Integrals can be negative. For instance, in the applet below, the function
f(x)=-3x^2+2x-2
is always negative, and so because of the caveat in the definition of the integral, the integral is negative so long as a is less than b. At the outset, a is 1 and b is 2, and the integral is -6. Adjust a and b, keeping a less than b, and notice that the integral is always negative.Quick Check: Use the above applet to calculate
Nuance 2: The numbers
a and b are called the bounds of the integral; a is called the lower (or left) bound, and b is called the right (or upper) bound. Almost always, it makes sense for a to be less than b, but it is possible to swap them, and have b less than a. If they are swapped, the effect is to switch the sign of the integral. For instance, if the integral was positive when a was less than b, then switching a and b will result in a negative integral. The opposite is also true. Try it out above by setting b to 1 and a to 2. This will make the integral positive 6 (it was negative 6 at the outset).Nuance 3: Never forget that the integral is a geometric concept that is heavily dependent on the geometric concept of "area". When the going gets challenging with integrals, 9 out of 10 times, it pays to come back to this geometric foundation of the integral.
Nuance 4: In a continuation of Nuance 3, also don't forget that integrals can be approximated by rectangles. We saw this in the first few activities of this chapter. This was not a one-off fluke; it's true of all integrals. For instance, in the applet below is both an integral and an approximation by rectangles. The rectangle sum always approximates the integral, and as the number of rectangles limits to infinity, the sum of the areas of the rectangles converges to the integral. Try it out! Increase
n (the number of rectangles), and observe that the rectangle sum approaches the numerical value of the integral. You might be interested to know that these approximating rectangles have a name: Riemann Sums. We won't study Riemann Sums much in this Season of Calculus for the People, but in Season 2, they'll come up a lot!There are some other nuances we'll discuss as they arise. Let's move on and discuss how to calculate integrals.