THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite.

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Presentation transcript:

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4

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 1.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. 2.When applying the Fundamental Theorem of Calculus, the following notation is convenient: 3.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

Assignment Assign: Page 293: Problems 6 – 20 e; 28,36 – 44e

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.

Example Find “c” for f(x) = 9/x 3 on [1, 3]. = -9/2x [-9/2(3) -2 ] – [-9/2(1) -2 ] [-1/2] – [-9/2] = 4 4 = f(c)(3 – 1) 4 = f(c)(2) 3 ∫9x- 3 dx 1

Example, continued… 4 = f(c)(2) 2 = 9/c 3 2c 3 = 9 c 3 = 9/2 c ≈ 1.650

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

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

Example Find F’(x) if F(x) = F’(x) = 1/x 3 dx x ∫1/t 3 dt 2

Assignment Page 293 & 295 Problems 46 – 54 evens AND 82 – 92 evens Omit # 48