The Art and Craft of Problem Solving

316 CHAPTER 9 CALCULUS

of y = f(x), the x-axis, and the vertical lines x = a and x = b. The right-hand side

has a completely different meaning, since it is related to f(x) by differentiation, the

computation of the slope of the tangent line to the graph of a function. How in the

world are areas and slopes related?

Stating it that way makes the FTC seem quite mysterious. Let us try to shed

some light on it. On one level, the FTC is an amazing algorithmic statement, since in

practice, antiderivatives are sometimes rather easy to compute. But that explains what

it is, not why it is true. Understanding why it is true is a matter of choosing the proper

interpretation of the entities in (1).

We start with the very useful define a function tool, which you have seen before

(for example, 5.4.2). Let

g(t) := l f(x)dx.

We chose the variable t on purpose, to make it easy to visualize g( t) as a function

of time. As t increases from a, the function g( t) is computing the area of a "moving

curtain," as seen below. Notice that g(a) = O.

y

y =/(x)

L-------------�---------L------�x

a

Differentiation is not just about tangent lines-it has a dynamic interpretation as in­

stantaneous rate of change. Thus g' (t) is equal to the rate of change of the area of the

curtain at time t. With this in mind, look at the picture below: what does your intuition

tell you the answer must be?

6.(

t+6.t

The area grows fast when the leading edge of the curtain is tall, and it grows slowly

when the leading edge is short. It makes intuitive sense that

g' (t) = f( t), (2)