# Fundamental Theorem of Calculus

• The two major branches of calculus have been introduced:

• differential calculus

• 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 slope of the secant line.

• Similarly, the area of a region under a curve was defined using 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.