Chapter 6 The Integral Sections 6.1, 6.2, and 6.3

The Integral The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area

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.

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.

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).

Antiderivatives However, the function G(x) = ⅓ x3 + 100 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?

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.

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

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.

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)

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

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).

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.

Power Rule for the Indefinite Integral Example:

Power Rule for the Indefinite Integral Indefinite Integral of ex and bx

Sum and Difference Rules Example:

Constant Multiple Rule Example:

Example - Different Variable Find the indefinite integral:

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

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

Integration by Substitution Example: Consider the integral: Sub to get Integrate Back Substitute

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

Example: Evaluate

Example: Evaluate

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.”

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.

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.”

Geometric Interpretation of the Definite Integral The Definite Integral As Area The Definite Integral As Net Change of Area

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

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

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

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

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

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

Area Using Geometry Example: Use geometry to compute the integral –1 5 Area = 8 Area = 2

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

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

R2 –1 1 5 R1 Total Area  2 + 8  10 Area of R1 Area of R2

Evaluating the Definite Integral Example: Calculate

Substitution for Definite Integrals Example: Calculate Notice limits change

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

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

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