Announcements Topics: -sections 7.3 (definite integrals), 7.4 (FTC), and 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

Riemann Sums and the Definite Integral Definition: The definite integral of a function on the interval from a to b is defined as a limit of the Riemann sum where is some sample point in the interval and

The Definite Integral Interpretation: If, then the definite integral is the area under the curve from a to b.

The Definite Integral Interpretation: If is both positive and negative, then the definite integral represents the NET or SIGNED area, i.e. the area above the x-axis and below the graph of f minus the area below the x-axis and above f

Definite Integrals and Area Example: Evaluate the following integrals by interpreting each in terms of area. (a)(b) (c)

Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

Summation Property of the Definite Integral (6) Suppose f(x) is continuous on the interval from a to b and that Then

Properties of the Definite Integral (7) Suppose f(x) is continuous on the interval from a to b and that Then

Types of Integrals Indefinite Integral Definite Integral antiderivative of f function of x number

The Fundamental Theorem of Calculus If is continuous on then where is any antiderivative of, i.e.,

Evaluating Definite Integrals Example: Evaluate each definite integral using the FTC. (a)(b) (c)(d)

Evaluating Definite Integrals Example: Try to evaluate the following definite integral using the FTC. What is the problem?

Differentiation and Integration as Inverse Processes If f is integrated and then differentiated, we arrive back at the original function f. If F is differentiated and then integrated, we arrive back at the original function F. FTC II FTC I

The Definite Integral - Total Change Interpretation: The definite integral represents the total amount of change during some period of time. Total change in F between times a and b: value at end value at start rate of change

Application – Total Change Example: Suppose that the growth rate of a fish is given by the differential equation where t is measured in years and L is measured in centimetres and the fish was 0.0 cm at age t=0 (time measured from fertilization).

Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

The Chain Rule and Integration by Substitution Recall: The chain rule for derivatives allows us to differentiate a composition of functions: derivative antiderivative

The Chain Rule and Integration by Substitution Suppose we have an integral of the form where Then, by reversing the chain rule for derivatives, we have composition of functions derivative of Inside function F is an antiderivative of f integrand is the result of differentiating a composition of functions

The Chain Rule and Integration by Substitution Suppose we have an integral of the form where Then, by reversing the chain rule for derivatives, we have composition of functions derivative of Inside function F is an antiderivative of f integrand is the result of differentiating a composition of functions

Example Integrate

Integration by Substitution Algorithm: 1.Let where is the part causing problems and cancels the remaining x terms in the integrand. 2.Substitute and into the integral to obtain an equivalent (easier!) integral all in terms of u.

Integration by Substitution Algorithm: 1.Let where is the part causing problems and cancels the remaining x terms in the integrand. 2.Substitute and into the integral to obtain an equivalent (easier!) integral all in terms of u.

Integration by Substitution Algorithm: 3. Integrate with respect to u, if possible. 4. Write final answer in terms of x again.

Integration by Substitution Algorithm: 3. Integrate with respect to u, if possible. 4. Write final answer in terms of x again.

Integration by Substitution Example: Integrate each using substitution. (a)(b) (c) (d)

The Product Rule and Integration by Parts The product rule for derivatives leads to a technique of integration that breaks a complicated integral into simpler parts. Integration by Parts Formula: hopefully this is a simpler Integral to evaluate given integral that we cannot solve

The Product Rule and Integration by Parts Deriving the Formula Start by writing out the Product Rule: Solve for

The Product Rule and Integration by Parts Deriving the Formula Start by writing out the Product Rule: Solve for

The Product Rule and Integration by Parts Deriving the Formula Integrate both sides with respect to x:

The Product Rule and Integration by Parts Deriving the Formula Simplify:

Integration by Parts Template: Choose: Compute: easy to integrate part part which gets simpler after differentiation

Integration by Parts Example: Integrate each using integration by parts. (a)(b) (c)(d)

Strategy for Integration MethodApplies when… Basic antiderivative…the integrand is recognized as the reversal of a differentiation formula, such as Guess-and-check…the integrand differs from a basic antiderivative in that “x” is replaced by “ax+b”, for example Substitution…both a function and its derivative (up to a constant) appear in the integrand, such as Integration by parts…the integrand is the product of a power of x and one of sin x, cos x, and e x, such as …the integrand contains a single function whose derivative we know, such as

Strategy for Integration What if the integrand does not have a formula for its antiderivative? Example: impossible to integrate

Approximating Functions with Polynomials Recall: The quadratic approximation to around the base point x=0 is base point

Integration Using Taylor Polynomials We approximate the function with an appropriate Taylor polynomial and then integrate this Taylor polynomial instead! Example: easy to integrate impossible to integrate for x-values near 0

Integration Using Taylor Polynomials We can obtain a better approximation by using a higher degree Taylor polynomial to represent the integrand. Example: