The Fundamental Theorem of Calculus Inverse Operations.

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Presentation transcript:

The Fundamental Theorem of Calculus Inverse Operations

Fundamental Theorem of Calculus Discovered independently by Gottfried Liebnitz and Isaac Newton Informally states that differentiation and definite integration are inverse operations.

Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], 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 used

Guidelines It is not necessary to include a constant of integration C in the antiderivative because they cancel out when you subtract.

Evaluating a Definite Integral

Evaluate the Definite Integral

Definite Integral Involving Absolute Value Evaluate

Definite Integral Involving Absolute Value

Using the Fundamental Theorem to Find Area Find the area of the region bounded by the graph of y = 2x 3 – 3x + 2, the x- axis, and the vertical lines x = 0 and x = 2

Using the Fundamental Theorem to Find Area

The Mean Value Theorem for Integrals If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that

Average Value of a Function This is just another way to write the Mean Value Theorem (mean = average in mathematics) If f is integrable on the closed interval [a,b], then the average value of f on the interval is

Average Value of a Function

Finding the Average Value of a Function Find the average value of f(x) = sin x on the interval [0,  ]

Force The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of sec x, where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is [0,  /3] and F(0) = 500.

Force (a) Find F as a function of x. F(x) = 500 sec 2 x (b) Find the average force exerted by the press over the interval [0,  /3]

Force

Second Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then, for every x in the interval,

Using the Second Fundamental Theorem of Calculus Evaluate

Second Fundamental Theorem of Calculus Find F’(x) of