Section 15.3 Area and Definite Integral

Slides:

Advertisements

Similar presentations

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.

Advertisements

Follow the link to the slide. Then click on the figure to play the animation. A Figure Figure

5/16/2015 Perkins AP Calculus AB Day 5 Section 4.2.

Applying the well known formula:

Riemann Sums. Objectives Students will be able to Calculate the area under a graph using approximation with rectangles. Calculate the area under a graph.

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

Drill Find the area between the x-axis and the graph of the function over the given interval: y = sinx over [0, π] y = 4x-x 3 over [0,3]

Warm Up Show all definite integrals!!!!! 1)Calculator Active: Let R be the region bounded by the graph of y = ln x and the line y = x – 2. Find the area.

Do Now: #10 on p.391 Cross section width: Cross section area: Volume:

Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:

4.2 Area Under a Curve.

David Loomis. is defined informally to be the signed area of the region in the xy-plane bounded by the graph of ƒ, the x- axis, and the vertical lines.

Aim: Riemann Sums & Definite Integrals Course: Calculus Do Now: Aim: What are Riemann Sums? Approximate the area under the curve y = 4 – x 2 for [-1, 1]

Integration 4 Copyright © Cengage Learning. All rights reserved.

Section 5.3 – The Definite Integral

Section 7.2a Area between curves.

CHAPTER 4 SECTION 4.2 AREA.

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

4.4 The Fundamental Theorem of Calculus

Brainstorm how you would find the shaded area below.

Warm Up 1) 2) 3)Approximate the area under the curve for 0 < t < 40, given by the data, above using a lower Reimann sum with 4 equal subintervals. 4)Approximate.

The Fundamental Theorem of Calculus

Section 4.3 Riemann Sums and Definite Integrals. To this point, anytime that we have used the integral symbol we have used it without any upper or lower.

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.

Sigma Notation, Upper and Lower Sums Area. Sigma Notation Definition – a concise notation for sums. This notation is called sigma notation because it.

11.5 Area After this lesson, you should be able to: Use sigma notation to write and evaluate a sum. Understand the concept of area. Approximate.

5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington.

1 Warm-Up a)Write the standard equation of a circle centered at the origin with a radius of 2. b)Write the equation for the top half of the circle. c)Write.

SECTION 4-2 (A) Application of the Integral. 1) The graph on the right, is of the equation How would you find the area of the shaded region?

Riemann Sums and Definite Integration y = 6 y = x ex: Estimate the area under the curve y = x from x = 0 to 3 using 3 subintervals and right endpoints,

4.2 Area Definition of Sigma Notation = 14.

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

Chapter 6 Integration Section 5 The Fundamental Theorem of Calculus (Day 1)

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.

Applications of Integration 7 Copyright © Cengage Learning. All rights reserved.

5.1 Areas and Distances. Area Estimation How can we estimate the area bounded by the curve y = x 2, the lines x = 1 and x = 3, and the x -axis? Let’s.

Definite Integrals & Riemann Sums

Definite Integrals. Definite Integral is known as a definite integral. It is evaluated using the following formula Otherwise known as the Fundamental.

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

Section 4.2 The Definite Integral. If f is a continuous function defined for a ≤ x ≤ b, we divide the interval [a, b] into n subintervals of equal width.

4.3 Riemann Sums and Definite Integrals

5.2 – The Definite Integral. Introduction Recall from the last section: Compute an area Try to find the distance traveled by an object.

1. Graph 2. Find the area between the above graph and the x-axis Find the area of each: 7.

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

Essential Question: How is a definite integral related to area ?

The Fundamental Theorem of Calculus Area and The Definite Integral OBJECTIVES  Evaluate a definite integral.  Find the area under a curve over a given.

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

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.

Calculus 4-R Unit 4 Integration Review Problems. Evaluate 6 1.

SECTION 4-3-B Area under the Curve. Def: The area under a curve bounded by f(x) and the x-axis and the lines x = a and x = b is given by Where and n is.

Application of the Integral

Area of a Region Between Two Curves (7.1)

Area Between the Curves

Area of a Region Between 2 Curves

Section 5.1 Area Between Curves.

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

Integration Review Problems

Finding the Area Between Curves

Area of a Region Between Two Curves (7.1)

Applying the well known formula:

AREA Section 4.2.

Definition: Sec 5.2: THE DEFINITE INTEGRAL

AP Calculus December 1, 2016 Mrs. Agnew

Section 5.3 – The Definite Integral

Section 5.3 – The Definite Integral

AREA Section 4.2.

Presentation transcript:

Section 15.3 Area and Definite Integral

Area Estimation How can we estimate the area bounded by the curve y = x2, the lines x = 1 and x = 3, and the x-axis? We will use 4 rectangles to approximate the area. Question How can we improve our estimation? http://science.kennesaw.edu/~plaval/applets/Riemann.html

Definition of the Area of a Region The area of the region bounded by the graph of f(x), the x-axis, the vertical lines x = a and x = b is where ci is the endpoint of the ith interval, and

The Definite Integral The definite integral of f from a to b is where a is called lower limit, b is called upper limit, If f(x) is above the x-axis, then the definite integral of f (x) from a to b represents the area of the region bounded by the graph of f(x), the x-axis , the vertical lines x = a and x = b.

Examples Evaluate the integrals by interpreting in terms of area.