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### Definite Integrals and Antiderivatives

Drill: Find dy/dx

- y = -cosx
- y = sin x
- y = ln (sec x)
- y = ln (sin x)

- dy/dx = sin x
- dy/dx = cos x
- dy/dx = (1/sec x)(tan x sec x) = tan x
- dy/dx = (1/sin x) (cos x) = cot x

Lesson 5.3

Objectives

- Students will be able to
- apply rules for definite integrals and find the average value of a function over a closed interval.

Rules for Definite Integrals

- Order of Integration
- Zero
- Constant Multiple

Rules for Definite Integrals

- Sum and Difference
- Additivity
- Max-Min Inequality: If max f and min f are the maximum and minimum values of f on [a, b], then

Rules for Definite Integrals

- Domination
f(x) > g(x) on [a,b]

f(x) > 0 on [a, b]

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

Example 1 Using the Rules for Definite Integrals

Suppose

Find each of the following integrals, if possible.

Not possible; not enough information given.

Average (Mean) Value

If f is integrable on the interval [a, b], the function’s average (mean) value on the interval is

Example 2 Applying the Definition of Average (Mean) Value

Find the average value of f (x) = 6 – x2 on [0, 5]. Where does f take on this value in the given interval?

Since 2.887 lies in the interval, the function does assume its average value in the interval.

Homework

- day 1: Page 290-292: 1-5 odd, 11-14, 47-49
- day 2: p. 291: 19-30, 31-35 odd

Using Antiderivativesfor Definite Integrals

If f is integrable over the interval [a, b], then

where f is the derivative of F.

Determining Integrals with Power Functions

Integrals: (where k and C are constants)

Note: when we are evaluating at definite integrals, we do not need to + C.

You will need to remember your derivative rules in order to do your anti-derivatives (integrals)

Example: If y = sin x, dy/dx = cos x

Therefore,

Example: if y = tan x, dy/dx = sec2x

Therefore,

I would strongly suggest that you dig out your derivatives’ sheet from chapter 3! (You may use it on your next quiz!)

Example 3 Finding an Integral Using do your anti-derivatives (integrals)Antiderivatives

Find each integral.

Example 3 Finding an Integral Using do your anti-derivatives (integrals)Antiderivatives

Find each integral.

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