AP Calculus Ms. Battaglia. If f and g are continuous on [a,b] and g(x)<f(x) for all x in [a,b], then the area of the region bounded by the graphs of f.

Slides:

Advertisements

Similar presentations

Miss Battaglia AB Calculus. Given a point, P, we want to define and calculate the slope of the line tangent to the graph at P. Definition of Tangent Line.

Advertisements

Integration: “the Definition of Area as a Limit; Sigma Notation”

CHAPTER Continuity Areas and Distances. The area problem: Find the area of the region S that lies under the curve y = f (x) from a to b. This means.

Miss Battaglia AP Calculus. Rotate around the x-axis and find the volume of y= 1 / 2 x from 0 to 4.

Ms. Battaglia AP Calculus. a) The terms of the sequence {a n } = {3 + (-1) n } are a) The terms of the sequence {b n } = are.

1 CALCULUS Even more graphing problems

MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.1 – Area Between Two Curves Copyright © 2006 by Ron Wallace, all.

Applications Of The Definite Integral The Area under the curve of a function The area between two curves.

AP Calculus Ms. Battaglia

Miss Battaglia AP Calculus. Let u be a differentiable function of x. 1.2.

Summary of area formula

4-3 DEFINITE INTEGRALS MS. BATTAGLIA – AP CALCULUS.

Gateway Arch, St. Louis, Missouri 6.1a Areas Between Curves.

Section 7.2a Area between curves.

Section 15.3 Area and Definite Integral

In this section, we will introduce the definite integral and begin looking at what it represents and how to calculate its value.

6/3/2016 Perkins AP Calculus AB Day 10 Section 4.4.

4.4 The Fundamental Theorem of Calculus

4-4 THE FUNDAMENTAL THEOREM OF CALCULUS MS. BATTAGLIA – AP CALCULUS.

Miss Battaglia AB/BC Calculus.  Connects differentiation and integration.  Integration & differentiation are inverse operations. If a function is continuous.

ESSENTIAL CALCULUS CH07 Applications of integration.

A PREVIEW OF CALCULUS Section 1.1. When you have finished your homework you should be able to… Understand what calculus is and how it compares with precalculus.

MAT 1236 Calculus III Section 15.3 Part II Double Integrals Over General Regions

AP CALCULUS AB Chapter 7: Applications of Definite Integrals Section 7.2: Areas in the Plane.

Estimating with Finite Sums

Miss Battaglia AB/BC Calculus.  Connects differentiation and integration.  Integration & differentiation are inverse operations. If a function is continuous.

Lecture III Indefinite integral. Definite integral.

The Fundamental Theorem of Calculus

1 Definite Integrals Section The Definite Integral The definite integral as the area of a region: If f is continuous and non-negative on the closed.

Area of a Region Between Two Curves

MAT 1235 Calculus II Section 5.1 Area Between Curves

MA Day 44 – March 14, 2013 Section 12.7: Triple Integrals.

Ms. Battaglia AP Calculus. Estimate y(4) with a step size h=1, where y(x) is the solution to the initial value problem: y’ – y = 0 ; y(0) = 1.

4.3 Riemann Sums and Definite Integrals. Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a.

Miss Battaglia AP Calculus. Let u be a differentiable function of x. 1.2.

AP CALCULUS AB CHAPTER 4, SECTION 2(ISH) Area 2013 – 2014 Revised

Area between curves AP Calculus Mrs. Mongold Gateway Arch, St. Louis, Missouri.

Volumes of Solids with Known Cross Sections

SECTION 7-3-C Volumes of Known Cross - Sections. Recall: Perpendicular to x – axis Perpendicular to y – axis.

4.3 Riemann Sums and Definite Integrals Definition of the Definite Integral If f is defined on the closed interval [a, b] and the limit of a Riemann sum.

Riemann sums & definite integrals (4.3) January 28th, 2015.

Calculus Sections 5.1 Apply exponential functions An exponential function takes the form y = a∙b x where b is the base and b>0 and b≠1. Identify as exponential.

A Preview of Calculus (1.1) September 18th, 2012.

Algebra 1 Section 7.6 Solve systems of linear inequalities The solution to a system of linear inequalities in two variable is a set of ordered pairs making.

4-3: Riemann Sums & Definite Integrals Objectives: Understand the connection between a Riemann Sum and a definite integral Learn properties of definite.

4.3 Finding Area Under A Curve Using Area Formulas Objective: Understand Riemann sums, evaluate a definite integral using limits and evaluate using properties.

C.1.5 – WORKING WITH DEFINITE INTEGRALS & FTC (PART 1) Calculus - Santowski 6/30/ Calculus - Santowski.

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite.

Algebra 2 cc Section 3.3 Relate zeros (roots), factors, and intercepts of polynomial functions Consider the quadratic function f(x) = x 2 – 2x - 8 Zeros.

Area Between Curves. Objective To find the area of a region between two curves using integration.

4-2 AREA AP CALCULUS – MS. BATTAGLIA. SIGMA NOTATION The sum of n terms a 1, a 2, a 3,…, a n is written as where i is the index of summation, a i is the.

EXAMPLE FORMULA DEFINITION 1.

Area of a Region Between Two Curves (7.1)

Algebra 2 Stat review 11.2, 11.3, 11.5.

Area and the Fundamental Theorem of Calculus

Section 5.1 Area Between Curves.

4-6 Numerical integration (The Trapezoidal Rule)

Warm-Up! Find the average value of

Area Between Polar Curves

Area of a Region Between Two Curves (7.1)

AP Calculus Honors Ms. Olifer

Section 7.2 Day 1 Disk method

Section 7.1 Day 1-2 Area of a Region Between Two Curves

4.2 Area Greenfield Village, Michigan

7.4 Slope Objectives: To count slope To use slope formula.

Section 7.2 Day 2 Disk Method

AP Calculus AB 8.2 Areas in the Plane.

Jay James Mike Norton Stephen Nesemann Kaitlyn Richardson

Riemann sums & definite integrals (4.3)

Presentation transcript:

AP Calculus Ms. Battaglia

If f and g are continuous on [a,b] and g(x)<f(x) for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x=a and x=b is

Set up the definite integral that gives the area of the region. y = x 2 -6x and y = 0

Set up the definite integral that gives the area of the region. y = x 2 +2x+1 and y = 2x+5

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region.

Known precalculus Formula Representative element New integration formula A=(height)(width) ΔA=[f(x)-g(x)]Δx For example, the area formula in this section was developed as follows:

 Page 454 #1,5,27, 31, 32, 33, 35, 48, 97, and Gasoline Worksheet