5.3 Definite Integrals. Example: Find the area under the curve from x = 1 to x = 2. The best we can do as of now is approximate with rectangles.

Slides:

Advertisements

Similar presentations

5.3 Definite Integrals and Antiderivatives Organ Pipe Cactus National Monument, Arizona Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

Advertisements

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

Riemann sums, the definite integral, integral as area

Applying the well known formula:

CHAPTER 4 THE DEFINITE INTEGRAL.

Chapter 5 Integrals 5.2 The Definite Integral In this handout: Riemann sum Definition of a definite integral Properties of the definite integral.

Antiderivatives and the Rules of Integration

Chapter 6 The Integral Sections 6.1, 6.2, and 6.3

Integration. Indefinite Integral Suppose we know that a graph has gradient –2, what is the equation of the graph? There are many possible equations for.

 Finding area of polygonal regions can be accomplished using area formulas for rectangles and triangles.  Finding area bounded by a curve is more challenging.

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

THE DEFINITE INTEGRAL RECTANGULAR APPROXIMATION, RIEMANN SUM, AND INTEGRTION RULES.

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

5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington.

Chapter 5 .3 Riemann Sums and Definite Integrals

5.3 Definite Integrals and Antiderivatives. 0 0.

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.

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,

Chapter 5 Integration Third big topic of calculus.

Section 4.3 – Riemann Sums and Definite Integrals

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

Antiderivatives: Think “undoing” derivatives Since: We say is the “antiderivative of.

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

1.5 Solving Inequalities Remember the rules of solving inequalities.

Time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3 ft/sec. Since rate. time = distance: If we.

5.2 Definite Integrals Bernhard Reimann

Area Under the Curve We want to approximate the area between a curve (y=x 2 +1) and the x-axis from x=0 to x=7 We want to approximate the area between.

Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics General Education Department Mathematics.

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.

BY DR. SAFA AHMED ELASKARY FACULTY OF ALLIED MEDICAL OF SCIENCES Lecture (1) Antiderivatives and the Rules of Integration.

Riemann Sums and The Definite Integral. time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3.

4.2 Area Definition of Sigma Notation = 14.

5.3 Definite Integrals and Antiderivatives. When I die, I want to go peacefully like my Grandfather did, in his sleep -- not screaming, like the passengers.

Integration – Overall Objectives  Integration as the inverse of differentiation  Definite and indefinite integrals  Area under the curve.

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

Adds the rectangles, where n is the number of partitions (rectangles) Height of rectangle for each of the x-values in the interval Width of partition:

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

Riemann Sum. When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. subinterval partition The width of a rectangle.

Definite Integrals & Riemann Sums

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.

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

30. Section 5.2 The Definite Integral Table of Contents.

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.

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.

Area Under the Curve We want to approximate the area between a curve (y=x2+1) and the x-axis from x=0 to x=7 We will use rectangles to do this. One way.

Indefinite Integrals or Antiderivatives

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

Section 6.2 Constructing Antiderivatives Analytically

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

Area and the Definite Integral

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

Area and the Definite Integral

The Area Question and the Integral

Sec 5.2: The Definite Integral

Lesson 3: Definite Integrals and Antiderivatives

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

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

Properties of Definite Integrals

5.3 Definite Integrals and Antiderivatives

1. Reversing the limits changes the sign. 2.

Riemann Sums and Integrals

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

Properties of Definite Integrals

6.2 Definite Integrals.

AP Calculus December 1, 2016 Mrs. Agnew

5.3 Definite Integrals and Antiderivatives MLK JR Birthplace

Presentation transcript:

5.3 Definite Integrals

Example: Find the area under the curve from x = 1 to x = 2. The best we can do as of now is approximate with rectangles

Example: Find the area under the curve from x = 1 to x = 2. The best we can do as of now is approximate with rectangles Let’s try 4 rectangles…  1.97

Integration Symbol lower limit of integration upper limit of integration integrand variable of integration So what do all of these symbols mean? Remember from last time:

Don’t forget that we are still finding the area under the curve. Now, how to evaluate the Definite Integral… Summation Symbol Rectangle Height Rectangle Base

And therefore that… Let F(x) be a function such that F'(x) = f(x) In other words, f is the derivative of F and F is the anti-derivative of f. From this we can also infer that… This will be useful…

xx Now let’s get the notation straight: …and therefore

xx And if we remove the limit (making it an approximation) Now let’s get the notation straight: and multiply both sides by  x, we get

   Since we are basing this on the left hand method, we stop at k – 1 Now let’s write out each of the rectangles… k = 1 k = 2 k = 3   

Now when we we add up all of the rectangles, what happens on the right side of the approximation?

The Definite Integral

Example: Find the area under the curve from x = 1 to x = 2. Area under the curve from x = 1 to x = 2. There can be a slight difference between finding the total area under the curve and finding the integral as we shall see next…

Example: Find the area between the x -axis and the curve from to. pos. neg. AHA! But here we need to remember that area under the x-axis will be negative. So let’s be careful what we are looking for.

Example: Find the definite integral of from to. pos. neg. If we simply wanted the integral from 0 to then we would just take one integral. = –1 – 0 =–1 …and notice that the lower area is –2 and the upper area is 1. so the integral is the sum of the two numbers not the sum of the actual areas.

Page 269 gives rules for working with integrals, the most important of which are: 2. If the upper and lower limits are equal, then the integral is zero. 1. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. 4. Integrals can be added and subtracted.

5. Intervals can be added (or subtracted.) 4. Integrals can be added and subtracted.