THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4

When you are done with your homework, you should be able to… Evaluate a definite integral using the Fundamental Theorem of Calculus Understand and use the Mean Value Theorem for Integrals Find the average value of a function over a closed interval Understand and use the Second Fundamental Theorem of Calculus  

Galileo lived in Italy from 1570-1642 Galileo lived in Italy from 1570-1642. He defined science as the quantitative description of nature—the study of time, distance and mass. He invented the 1st accurate clock and telescope. Name one of his advances. He discovered laws of motion for a falling object. He defined science. He formulated the language of physics.. All of the above.

THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite integrals are inverse operations The slope of the tangent line was defined using the quotient The area of a region under a curve was defined using the product The Fundamental Theorem of Calculus states that the limit processes used to define the derivative and definite integral preserve this relationship

Theorem: The Fundamental Theorem of Calculus If a function f is continuous on the closed interval and F is an antiderivative of f on the interval , then

Guidelines for Using the Fundamental Theorem of Calculus Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. When applying the Fundamental Theorem of Calculus, the following notation is convenient: It is not necessary to include a constant of integration in the antiderivative because

Example: Find the area of the region bounded by the graph of , the x-axis, and the vertical lines and .

Find the area under the curve bounded by the graph of , , and the x-axis and the y-axis. 9/4 0.0

THE MEAN VALUE THEOREM FOR INTEGRALS If f is continuous on the closed interval , then there exists a number c in the closed interval such that  

So…what does this mean?! Somewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area of the region under the curve.

AVERAGE VALUE OF A FUNCTION If f is integrable on the closed interval , then the average value of f on the interval is

Find the average value of the function 13.0 0.0

THE SECOND FUNDAMENTAL THEOREM OF CALCULUS If f is continuous on an open interval I containing c, then, for every x in the interval,