If a < b < c, then for any number b between a and c, the integral from a to c is the integral from a to b plus the integral from b to c. Theorem: Section 4.4 – Properties of Definite Integrals

Example:

Section 4.4 – Properties of Definite Integrals Example:

Section 4.4 – Properties of Definite Integrals Copyright  2010 Pearson Education, Inc. As the number of rectangles increased, the approximation of the area under the curve approaches a value.

Copyright  2010 Pearson Education, Inc. Section 4.4 – Properties of Definite Integrals

Example: Section 4.4 – Properties of Definite Integrals

Example: Section 4.4 – Properties of Definite Integrals Find the points of intersection

Average Value of a Continuous Function Copyright  2010 Pearson Education, Inc. Section 4.4 – Properties of Definite Integrals

Average Value of a Continuous Function Section 4.4 – Properties of Definite Integrals

a) Find the total profit from the first 10 days. b) Find the average daily profit from the first 10 days. Reminder: a)

Section 4.4 – Properties of Definite Integrals a) Find the total profit from the first 10 days. b) Find the average daily profit from the first 10 days. Reminder: b)

Section 4.4 – Properties of Definite Integrals

Differentiation Review: Copyright  2010 Pearson Education, Inc. Integration: Section 4.5 – Integration Techniques: Substitution

Copyright  2010 Pearson Education, Inc. Integration: Section 4.5 – Integration Techniques: Substitution

Copyright  2010 Pearson Education, Inc. Integrate: Section 4.5 – Integration Techniques: Substitution

Integrate: Section 4.5 – Integration Techniques: Substitution

Copyright  2010 Pearson Education, Inc. Integrate: Section 4.5 – Integration Techniques: Substitution

Integrate: Section 4.5 – Integration Techniques: Substitution

Copyright  2010 Pearson Education, Inc. Integrate: Section 4.5 – Integration Techniques: Substitution

Integrate: Section 4.5 – Integration Techniques: Substitution

Section 4.4 – Properties of Definite Integrals