Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions.

Slides:

Advertisements

Similar presentations

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

Advertisements

6.5 The Definite Integral In our definition of net signed area, we assumed that for each positive number n, the Interval [a, b] was subdivided into n subintervals.

Applying the well known formula:

CHAPTER 4 THE DEFINITE INTEGRAL.

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

Area Between Two Curves

Area Between Two Curves

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Definite Integrals Finding areas using the Fundamental Theorem of Calculus.

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

§12.5 The Fundamental Theorem of Calculus

4.6 Numerical Integration -Trapezoidal Rule -Simpson’s Rule

Trapezoidal Approximation Objective: To find area using trapezoids.

The Integral chapter 5 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental.

Chapter 5 .3 Riemann Sums and Definite Integrals

Georg Friedrich Bernhard Riemann

6.3 Definite Integrals and the Fundamental Theorem.

First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.

State Standard – 16.0a Students use definite integrals in problems involving area. Objective – To be able to use the 2 nd derivative test to find concavity.

7.4: The Fundamental Theorem of Calculus Objectives: To use the FTC to evaluate definite integrals To calculate total area under a curve using FTC and.

5.2 Definite Integrals. Subintervals are often denoted by  x because they represent the change in x …but you all know this at least from chemistry class,

5.2 Definite Integrals.

If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation.

Section 5.2: Definite Integrals

Chapter 6 The Definite Integral. § 6.1 Antidifferentiation.

Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice.

The Fundamental Theorem of Calculus (4.4) February 4th, 2013.

4.4 The Fundamental Theorem of Calculus

Fundamental Theorem of Calculus: Makes a connection between Indefinite Integrals (Antiderivatives) and Definite Integrals (“Area”) Historically, indefinite.

Warm Up – NO CALCULATOR Let f(x) = x2 – 2x.

5.4 Fundamental Theorem of Calculus Quick Review.

The Fundamental Theorem of Calculus Objective: The use the Fundamental Theorem of Calculus to evaluate Definite Integrals.

Warm-Up: (let h be measured in feet) h(t) = -5t2 + 20t + 15

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

Antidifferentiation: The Indefinite Intergral Chapter Five.

Chapter 5 – The Definite Integral. 5.1 Estimating with Finite Sums Example Finding Distance Traveled when Velocity Varies.

MAT 212 Brief Calculus Section 5.4 The Definite Integral.

The Fundamental Theorem of Calculus

5.2 Definite Integrals Bernhard Reimann

Chapter 6 INTEGRATION An overview of the area problem The indefinite integral Integration by substitution The definition of area as a limit; sigma notation.

5.2 Definite Integrals. When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. subinterval partition The width.

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

4032 Fundamental Theorem AP Calculus. Where we have come. Calculus I: Rate of Change Function.

4.2 Copyright © 2014 Pearson Education, Inc. Antiderivatives as Areas OBJECTIVE Find the area under a graph to solve real- world problems Use rectangles.

Fundamental Theorem AP Calculus. Where we have come. Calculus I: Rate of Change Function.

Areas & Riemann Sums TS: Making decisions after reflection and review.

Ch. 6 – The Definite Integral

Warm up Problems More With Integrals It can be helpful to guess and adjust Ex.

The Fundamental Theorem of Calculus is appropriately named because it establishes connection between the two branches of calculus: differential calculus.

To find the area under the curve Warm-Up: Graph. Area under a curve for [0, 3]  The area between the x-axis and the function Warm-up What is the area.

AP CALC: CHAPTER 5 THE BEGINNING OF INTEGRAL FUN….

5.2 Definite Integrals Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

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

“In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used.

5.3 Definite Integrals and Riemann Sums. I. Rules for Definite Integrals.

Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice.

4.3: Definite Integrals Learning Goals Express the area under a curve as a definite integral and as limit of Riemann sums Compute the exact area under.

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

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Integration 5 Antiderivatives Substitution Area Definite Integrals Applications.

Definite Integrals, The Fundamental Theorem of Calculus Parts 1 and 2 And the Mean Value Theorem for Integrals.

Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice.

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

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

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

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Riemann Sums and Integrals

6.2 Definite Integrals.

More Definite Integrals

(Finding area using integration)

Presentation transcript:

Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions

Objectives  To develop a formula for finding the area under a curve.

Area Under the Curve How do we find areas under a curve, but above the x-axis?

Area Under the Curve As the number of rectangles used to approximate the area of the region increases, the approximation becomes more accurate.

Area Under the Curve It is possible to find the exact area by letting the width of each rectangle approach zero. Doing this generates an infinite number of rectangles.

Area Under the Curve Find the area of the shaded region.

Area Under the Curve = um (height)  (base) Area  sum of the areas of the rectangles = = a b The formula looks like an integral.

Area Under the Curve = a b The formula looks like an integral. Is the area really given by the antiderivative? Yes! The definite integral of f from a to b is the limit of the Riemann sum as the lengths of the subintervals approach zero.

Area Under the Curve A function and the equation for the area between its graph and the x-axis are related to each other by the antiderivative. = a b The formula looks like an integral.

Two Questions of Calculus Q 1 : How do you find instantaneous velocity? A: Use the derivative. Q 2 : How do you find the area of exotic shapes? A: Use the antiderivative.

Area Under the Curve How do we calculate areas under a curve, but above the x-axis?

The Fundamental Theorem of Calculus

Find the area between the graph of f and the x-axis on the interval [0, 3].

The Fundamental Theorem of Calculus Find the area between the graph of f and the x-axis on the interval [0, 3]. The bar tells you to evaluate the expression at 3 and subtract the value of the expression at 0.

The Fundamental Theorem of Calculus

Find the area between the graph of f and the x-axis on the interval [0, 1].

The Fundamental Theorem of Calculus Substitute into the integral. Always express your answer in terms of the original variable.

The Fundamental Theorem of Calculus So this would represent the area between the curve y = x √(x 2 + 1) and the x-axis from x = 0 to 1

Conclusion   As the number of rectangles used to approximate the area of the region increases, the approximation becomes more accurate.   It is possible to find the exact area by letting the width of each rectangle approach zero. Doing this generates an infinite number of rectangles.   A function and the equation for the area between its graph and the x-axis are related to each other by the antiderivative.   The Fundamental Theorem of Calculus enables us to evaluate definite integrals. This empowers us to find the area between a curve and the x-axis.