Introduction

In this section we shall state five basic theorems about the integral, culminating in the Fundamental Theorem of Calculus. Right now we can only approximate a definite integral by the laborious computation of a finite Riemann sum. At the end of this section we will be in a position easily to compute exact values for many definite integrals. The key to the method is the Fundamental Theorem. Our first theorem shows that we are free to choose any positive infinitesimal we wish for dx in the definite integral.

• Theorem 1: infintesimals can be arbitrary
• Theorem 2: the rectangle property
• Theorem 3: the addition property
• Theorem 4: continuous integral
• Theorem 5: fundamental theorem of calculus