FTC简介

So far, we've learned about differentiation and integration. So a natural question to ask is: are those two topics related?

This question is answered by the two Fundamental Theorems of Calculus. In essence, the first theorem states that every function has an antiderivative, which can be found by integration. On the other hand, the second theorem states that the definite integral of a function can be calculated by using any of the function's antiderivatives.

(Sometimes the two theorems are labeled the other way around.)

To find out more, click on the next sections!

Before Newton and Leibniz invented calculus, people had already known how to find the tangent of a curve or to compute the area under a curve for some specific cases. For example, Archimedes had already successfully calculated the area under a parabola using the method of exhaustion. However, it is Newton and Leibniz' work to discover the inverse relationship between differentiation and integration.

Briefly, what are the fundamental theorems of calculus (FTC), and why are they important? The FTC—essentially an inverse relationship between differentiation and integration—allows you to compute integrals by anti-differentiation, which can be useful in many applications of integrals. And just like the inverse relationship between addition and subtraction allows you to solve the addition equation $x+2 = 5$ by subtraction, the FTC allows you to solve some differential equations using integration, and to transform many others into integral equations.