Estimating with Finite Sums

Quick Review Slide 5- 2

Quick Review Solutions Slide 5- 3

What you’ll learn about Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output … and why Learning about estimating with finite sums sets the foundation for understanding integral calculus. Slide 5- 4

Example Finding Distance Traveled when Velocity Varies Slide 5- 5

Example Finding Distance Traveled when Velocity Varies Slide 5- 6

LRAM, MRAM, and RRAM approximations to the area under the graph of y=x2 from x=0 to x=3 Slide 5- 7

Example Estimating Area Under the Graph of a Nonnegative Function Slide 5- 8

5.2 Definite Integrals

Quick Review Slide 5- 10

Quick Review Solutions Slide 5- 11

What you’ll learn about Riemann Sums The Definite Integral Computing Definite Integrals on a Calculator Integrability … and why The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus. Slide 5- 12

Sigma Notation Slide 5- 13

The Definite Integral as a Limit of Riemann Sums Slide 5- 14

The Existence of Definite Integrals Slide 5- 15

The Definite Integral of a Continuous Function on [a,b] Slide 5- 16

The Definite Integral Slide 5- 17

Example Using the Notation Slide 5- 18

Area Under a Curve (as a Definite Integral) Slide 5- 19

Area Slide 5- 20

The Integral of a Constant Slide 5- 21

Example Using NINT Slide 5- 22

Definite Integrals and Antiderivatives 5.3 Definite Integrals and Antiderivatives

Quick Review Slide 5- 24

Quick Review Solutions Slide 5- 25

What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for Definite Integrals Connecting Differential and Integral Calculus … and why Working with the properties of definite integrals helps us to understand better the definite integral. Connecting derivatives and definite integrals sets the stage for the Fundamental Theorem of Calculus. Slide 5- 26

Rules for Definite Integrals Slide 5- 27

Example Using the Rules for Definite Integrals Slide 5- 28

Example Using the Rules for Definite Integrals Slide 5- 29

Example Using the Rules for Definite Integrals Slide 5- 30

Average (Mean) Value Slide 5- 31

Example Applying the Definition Slide 5- 32

The Mean Value Theorem for Definite Integrals Slide 5- 33

The Mean Value Theorem for Definite Integrals Slide 5- 34

The Derivative of an Integral Slide 5- 35

Quick Quiz Sections 5.1 - 5.3 Slide 5- 36

Quick Quiz Sections 5.1 - 5.3 Slide 5- 37

Quick Quiz Sections 5.1 - 5.3 Slide 5- 38

Quick Quiz Sections 5.1 - 5.3 Slide 5- 39

Quick Quiz Sections 5.1 - 5.3 Slide 5- 40

Quick Quiz Sections 5.1 - 5.3 Slide 5- 41