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7.4: The Fundamental Theorem of Calculus Objectives: To use the FTC to evaluate definite integrals To calculate total area under a curve using FTC and geometric formulas To connect the derivative and the integral using FTC

Properties of the Definite Integral (some repeated from 7.1) If all indicated definite integrals exist, then: 1. 2. for any real # k

Properties cont… 3. 4. for any real # c

Properties cont… 5. 6.If f(x) < g(x) for all x, then

And last one… 7., where c is a constant

Using the Properties Given: Evaluate: a.) b.) c.) d.)

Evaluate using properties and area:

Find if:

When evaluating the definite integral using area under the curve, subtract areas that fall below the x-axis:

Fundamental Theorem of Calculus, Part 1 Let What is A(0), A(1), and A(2)? Find a general formula for A(x) and A’ (x).

Fundamental Theorem of Calculus, Part 1  Connects the integral and the derivative Suppose f(t) is a continuous function on some interval [a,b]. Let Then A(x) is differentiable and A’(x) = f(x) (The derivative of the integral function is the integrand with a change in variable)

Example Find y’.

Fundamental Theorem of Calculus, Part 2  Involves the antiderivative  Shows how to evaluate the definite integral directly from antiderivatives Let f be continuous on [a,b], and let F be any antiderivative of f. Then

How to use FTC part 2: (basic problems) 1.Find antiderivative of integrand 2.Evaluate the antiderivative for the upper limit, and subtract the antiderivative of the lower limit  MUST BE CONTINUOUS ON INTERVAL!!  FTC2 does have its limitations…not all integrals can be evaluated using it (can’t find antiderivative).

Example: Evaluate:

Examples: 1. 2. 3.

Example:

Examples:Evaluate 1. 2. 3.

Revisiting FTC part 1…why it works! 1. Find F ‘(x) 2. Find F’(x)

Can you use FTC part 2 to evaluate the following?

Using Substitution Method 1: Integrating in terms of u, need to change upper and lower bounds Evaluate:

Method 2: Find indefinite integral first, then evaluate using original limits

Example The rate at which a substance grows is given by where x is the time (in days). What is the total accumulated growth during the first 3.5 days?

Finding Total Area  The definite integral is a number; it accounts for regions of curve below the x-axis  When you use FTC 2, area below the x-axis is subtracted To find TOTAL AREA between the graph y=f(x) and the x-axis over the interval [a,b]: 1.Partition [a,b] by finding the zeros of f. 2.Integrate f over each subinterval 3.Add the absolute values of the integrals

Find the area of the region between the curve y=9-x 2 and the x-axis over the interval [0,4]

Connecting the ideas… Graph the function over the interval. a.) Integrate the function over interval b.) Find the total area of the region between the graph and the x-axis y = x 2 -6x+8. [0, 3]

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