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**Chapter 6 The Integral Sections 6.1, 6.2, and 6.3**

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**The Integral The Indefinite Integral Substitution**

The Definite Integral As a Sum The Definite Integral As Area

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Introduction A physicist who knows the velocity of a particle might wish to know its position at a given time. A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time.

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**Antiderivatives Definition**

In each case, the problem is to find a function F whose derivative is a known function f. If such a function F exists, it is called an antiderivative of f. Definition A function F is called an antiderivative of f on an interval I if F’(x) = f (x) for all x in I.

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**Antiderivatives For instance, let f (x) = x2.**

It is not difficult to discover an antiderivative of f if we keep the Power Rule in mind. In fact, if F(x) = ⅓ x3, then F’(x) = x2 = f (x).

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Antiderivatives However, the function G(x) = ⅓ x also satisfies G’(x) = x2. Therefore, both F and G are antiderivatives of f. Indeed, any function of the form H(x)=⅓ x3 + C, where C is a constant, is an antiderivative of f. The question arises: Are there any others?

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Antiderivatives Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C where C is an arbitrary constant.

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Antiderivatives Going back to the function f (x) = x2, we see that the general antiderivative of f is ⅓ x3 + C.

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Family of Functions By assigning specific values to C, we obtain a family of functions. Their graphs are vertical translates of one another. This makes sense, as each curve must have the same slope at any given value of x.

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**Notation for Antiderivatives**

The symbol is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f . Thus, means F’(x) = f (x)

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**Indefinite Integral The expression:**

read “the indefinite integral of f with respect to x,” means to find the set of all antiderivatives of f. x is called the variable of integration Integrand Integral sign

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**Indefinite Integral For example, we can write**

Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).

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**Constant of Integration**

Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant. Example: Represents every possible antiderivative of 6x.

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**Power Rule for the Indefinite Integral**

Example:

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**Power Rule for the Indefinite Integral**

Indefinite Integral of ex and bx

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**Sum and Difference Rules**

Example:

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**Constant Multiple Rule**

Example:

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**Example - Different Variable**

Find the indefinite integral:

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**Position, Velocity, and Acceleration Derivative Form**

If s = s(t) is the position function of an object at time t, then Velocity = v = Acceleration = a = Integral Form

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**Integration by Substitution**

Method of integration related to chain rule. If u is a function of x, then we can use the formula

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**Integration by Substitution**

Example: Consider the integral: Sub to get Integrate Back Substitute

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Example: Evaluate Pick u, compute du Sub in Integrate Sub in

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Example: Evaluate

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Example: Evaluate

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The Definite Integral Let f be a continuous function on [a, b]. If F is any antiderivative of f defined on [a, b], then the definite integral of f from a to b is defined by is read “the integral, from a to b of f (x) dx.”

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**Notation In the notation , f (x) is called the integrand.**

a and b are called the limits of integration; a is the lower limit and b is the upper limit. For now, the symbol dx has no meaning by itself; is all one symbol. The dx simply indicates that the independent variable is x.

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The Definite Integral The procedure of calculating an integral is called integration. The definite integral is a number. It does not depend on x. Also note that the variable x is a “dummy variable.”

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**Geometric Interpretation of the Definite Integral**

The Definite Integral As Area The Definite Integral As Net Change of Area

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**Definite Integral As Area**

If f is a positive function defined for a ≤ x ≤ b, then the definite integral represents the area under the curve y = f (x) from a to b

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**Definite Integral As Area**

If f is a negative function for a ≤ x ≤ b, then the area between the curve y = f (x) and the x-axis from a to b, is the negative of

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**Definite Integral As Area**

Consider y = f (x) = 0.5x + 6 on the interval [2,6] whose graph is given below,

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**Definite Integral As Area**

Consider y = f (x) = 0.5x + 6 on the interval [2,6] whose graph is given below,

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**Definite Integral as Net Area**

If f changes sign on the interval a ≤ x ≤ b, then definite integral represents the net area, that is, a difference of areas as indicated below: a b R1 R2 R3 Area of R1 – Area of R2 + Area of R3

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Total Area If f changes sign on the interval a ≤ x ≤ b, and we need to find the total area between the graph and the x-axis from a to b, then Total Area Area of R1 + Area of R2 + Area of R3 Area of R1 a b R1 R2 R3 Area of R2 c d Area of R3

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**Area Using Geometry Example: Use geometry to compute the integral**

–1 5 Area = 8 Area = 2

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**Area Using Antiderivatives**

Example: Use an antiderivative to compute the integral First, we need an antiderivative of

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**Area Using Antiderivatives**

Example: Now find the total area bounded by the curve and the x-axis from x –1 to x 5. R2 –1 1 5 R1 Total Area Area of R1 + Area of R2

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R2 –1 1 5 R1 Total Area 10 Area of R1 Area of R2

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**Evaluating the Definite Integral**

Example: Calculate

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**Substitution for Definite Integrals**

Example: Calculate Notice limits change

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Computing Area Example: Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of Gives the area since 2x3 is nonnegative on [0, 2]. Antiderivative

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**The Definite Integral As a Total**

If r (x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

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**The Definite Integral As a Total**

Example: If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

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