Fundamental Theorem of Calculus

The two major branches of calculus have been introduced: differential calculus and integral calculus. At this point, these two problems might seem unrelated—but there is a very close connection. The connection was discovered independently by Isaac Newton and Gottfried Leibniz and is stated in a theorem that is appropriately called the Fundamental Theorem of Calculus.

Informally, the theorem states that differentiation and (definite) integration are inverse operations, in the same sense that division and multiplication are inverse operations. The slope of the tangent line was defined using the (the slope of the secant line). Similarly, the area of a region under a curve was defined using the (the area of a rectangle).

So, at least in the primitive approximation stage, the operations of differentiation and definite integration appear to have an inverse relationship in the same sense that division and multiplication are inverse operations. The Fundamental Theorem of Calculus states that the limit processes (used to define the derivative and definite integral) preserve this inverse relationship.

Essential Questions

How are definite integrals evaluated using the Fundamental Theorem of Calculus?