The Area Under the Bell Curve Our last example of a deﬁnite integral is: � x 2 F (x) = e−t dt. 0 As we’ve already remarked, this is a new function that we can’t express in terms of functions we already know. To begin to understand the properties of this function F we’ll sketch its graph. area F (x) x 2 Figure 1: Area under e−t . The fundamental theorem tells us right away that: 2 F � (x) = e−x . It’s easy to ﬁnd the value of F (0), because the “starting place” of the integral is 0. � 0 2 F (0) = e−t dt = 0. 0 We can also compute the second derivative: F �� (x) = −2xe−x 2 The ﬁrst derivative is always positive, so F (x) is always increasing. Because 2 the sign of −2xe−x is just the sign of −2x, its graph will be concave down when 2 x > 0 and concave up when x < 0. And ﬁnally, F � (0) = e−0 = 1. (We often deﬁne functions so that their graphs have slope 1 in convenient locations.) By combining all this information we get a pretty good idea of what the graph of the function looks like, even though we cannot write down a strictly algebraic equation for F (x). We want to know as much as possible about this function, so we’ll discuss a few more features before moving on. First we show that F (x) is odd. As mentioned in Figure 1, F (x) is the area 2 under the graph of e−t between 0 and x. From the symmetry of the graph we see: � � 0 x 2 e−x dx = −x 2 e−x dx, 0 so: � F (−x) = 0 −x 2 e−x dx = − � 0 −x 2 e−x dx = − � 0 x 2 e−x dx = −F (x). Figure 2: Graph of F (x) = �x 0 2 e−t dt. We conclude that F (x) is an odd function. Because F is odd we know that the part of its graph to the right of the y-axis is exactly a rotation of the part of its graph to the left of the y-axis. If we know the shape of one branch we immediately know the shape of the other branch. Our ﬁnal step to understanding the graph is to ﬁgure out what happens at the ends. It turns out that the graph approaches a horizontal asymptote as x approaches positive inﬁnite and, by symmetry, as x goes to negative inﬁnity. On the right, the graph rises to a certain level; on the left it falls to the negative of that level. 2 What is that level? It’s the area under the graph of f (x) = e−t between 0 √ and inﬁnity: 2π . It took several years for mathematicians to discover the exact value of this area. √ π lim F (x) = x→∞ 2 √ π lim F (x) = − x→−∞ 2 √ Because 1 is a nicer number to work with than 2π , people deﬁne the error function to be: � x 2 2 2 erf(x) = √ e−t dt = √ F (x). π 0 π This is a famous function, as is its cousin the standard normal distribution.

The Area Under the Bell Curve

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