The need to calculate instantaneous rate of change led the discoveries of calculus to an investigation of the slopes of tangent lines and, ultimately, to the derivative--to what is called differential calculus. But derivatives revealed only half the story. In addition to a calculation method (a "calculus") to describe how functions change at any given instant, they needed a method to describe how those instantaneous changes could accumulate over an interval to produce the function that describes the total change.
Early in the 14th century, a group of scholars in Oxford, England, explored how to find the accumulated distance from knowledge of the velocity. In 1638, Galileo used their ideas to explain the motion of falling bodies and justify his claim that the earth circles the sun. Later in that century, Isaac Newton studied these accumulation functions, and in 1666, discovered a remarkable insight that connected accumulation functions to what he already knew about derivatives. If there was one moment when calculus was born, this was it.
Today, what we call the integral calculus or integration has two distinct interpretations. We begin this chapter by looking at integration as accumulation. but is also can be viewed as reversing the process of differentiation, what we call antidifferentiation. Newton's insight, that these two are connected, is what is called the Fundamental Theorem of Calculus. (Taken from an excerpt in the fifth edition of the Finney, Demana, Waits, Kennedy, and Bressoud textbook Calculus - Graphical, Numerical, Algebraic.)
Early in the 14th century, a group of scholars in Oxford, England, explored how to find the accumulated distance from knowledge of the velocity. In 1638, Galileo used their ideas to explain the motion of falling bodies and justify his claim that the earth circles the sun. Later in that century, Isaac Newton studied these accumulation functions, and in 1666, discovered a remarkable insight that connected accumulation functions to what he already knew about derivatives. If there was one moment when calculus was born, this was it.
Today, what we call the integral calculus or integration has two distinct interpretations. We begin this chapter by looking at integration as accumulation. but is also can be viewed as reversing the process of differentiation, what we call antidifferentiation. Newton's insight, that these two are connected, is what is called the Fundamental Theorem of Calculus. (Taken from an excerpt in the fifth edition of the Finney, Demana, Waits, Kennedy, and Bressoud textbook Calculus - Graphical, Numerical, Algebraic.)