Theorem 4: Continuous Integral

SKETCH OF PROOF

Let c be in I, and let x be infinitely close to c and between the endpoints of I. By the Addition Property,

and

This is the area of the infinitely thin strip under the curve y = f(t) between t = x and t = c (see Figure 4.2.8). The strip has width Δx = c - x. By the Rectangle Property, its area is between m Δx and M Δx and hence is infinitely small. Therefore F(x) is infinitely close to F(c), and F is continuous on I.

Figure 4.2.8

Our fifth theorem, the Fundamental Theorem of Calculus, shows that the definite integral can be evaluated by means of antiderivatives. The process of antidifferentiation is just the opposite of differentiation. To keep things simple, let I be an open interval, and assume that all functions mentioned have domain I.

For example, suppose a particle is moving upward along the y-axis with velocity v = f(t) and position y = F(t) at time t. The position y = F(t) is an anti-derivative of the velocity v = f(t). We shall discuss antiderivatives in more detail in the next section. We are now ready for the Fundamental Theorem.