Essential Question: How is a definite integral related to area ?

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Presentation transcript:

Essential Question: How is a definite integral related to area ?

Definition of a Definite Integral A definite integral of a function y = f(x) on an interval [a, b] is the signed (  ) area between the curve and the x-axis.

Evaluating a Definite Integral (The Fundamental Theorem of Calculus, Part 1) If f is continuous on [a, b] and F(x) is any antiderivative of f(x), then You first take the antiderivative F(x) and then subtract the values at each endpoint (upper – lower)

ANTIDIFFERENTIATION

The following functions f(x), represent derivatives of some other functions F(x). Reverse the power rule process and find the functions F(x).

THE POWER RULE For any rational power r  -1, Notice, you add 1 first, then divide! (We will come back to the “c” later!)

ANTIDERIVATIVE OF A CONSTANT For any constant ‘k’,

Evaluate each definite integral

Ex 4: Find the area of the region between the graph of f(x) = x 2 – 4 and the x-axis from x = -2 to x = 2

Ex 5: Find the area of the region between the graph of f(x) = -x 2 + 4x – 3 and the x-axis from x = 0 to x = 2 The function crosses the x-axis within the given interval! (at x = 1)

Solution Ex 4: TOTAL AREA: 2 SQUARE UNITS

TRIG ANTIDERIVATIVES

LOG AND EXPONENTIAL

Constant Rule #1 Suppose that f(x) has an antiderivative. Then, for any constants a and b, You can ‘pull a constant out’ of an integral This is useful when your constant is something like 

Evaluate each definite integral

What’s up with the “c”?

Find the derivative of the following functions Why do we add “c”?

SIMPLE EXAMPLES: Evaluate each integral.

RANDOM EXAMPLES: Evaluate each integral.