# 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.