Lesson 16 and 17 Area and Riemann Sums

Slides:

Advertisements

Similar presentations

An Introduction to Calculus. Calculus Study of how things change Allows us to model real-life situations very accurately.

Advertisements

Numerical Integration

Section 8.5 Riemann Sums and the Definite Integral.

Created for ENMU Tutoring/Supplemental InstructionPeterson Fall 2011.

A Riemann sum is a method for approximating the total area underneath a curve on a graph. This method is also known as taking an integral. There are 3.

Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4.

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

Using Areas to Approximate to Sums of Series In the previous examples it was shown that an area can be represented by the sum of a series. Conversely the.

1 Example 2 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,  ] into 6 subintervals. Solution Observe that the function.

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:

Every slope is a derivative. Velocity = slope of the tangent line to a position vs. time graph Acceleration = slope of the velocity vs. time graph How.

Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

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.

1 §12.4 The Definite Integral The student will learn about the area under a curve defining the definite integral.

5.1 Estimating with Finite Sums Greenfield Village, Michigan.

Problem of the Day If f ''(x) = x(x + 1)(x - 2) 2, then the graph of f has inflection points when x = a) -1 onlyb) 2 only c) -1 and 0 onlyd) -1 and 2.

Section 5.1/5.2: Areas and Distances – the Definite Integral Practice HW from Stewart Textbook (not to hand in) p. 352 # 3, 5, 9 p. 364 # 1, 3, 9-15 odd,

Chapter 5: Calculus~Hughes-Hallett §The Definite Integral.

A Preview of Calculus Lesson 1.1. What Is Calculus It is the mathematics of change It is the mathematics of –tangent lines –slopes –areas –volumes It.

Warm Up – Calculator Active

1 Example 1 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,1] into 4 subintervals. Solution This definite integral.

Estimating area under a curve

Observations about Error in Integral Approximations The Simplest Geometry.

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.

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,

Section 4.3 Day 1 Riemann Sums and Definite Integrals AP Calculus BC.

Integral Review Megan Bryant 4/28/2015. Bernhard Riemann  Bernhard Riemann ( ) was an influential mathematician who studied under Gauss at the.

In this section, we will begin to look at Σ notation and how it can be used to represent Riemann sums (rectangle approximations) of definite integrals.

WS: Riemann Sums. TEST TOPICS: Area and Definite Integration Find area under a curve by the limit definition. Given a picture, set up an integral to calculate.

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

Section 4.3 Day 2 Riemann Sums & Definite Integrals AP Calculus BC.

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

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

Area under a curve To the x axis.

The Definite Integral. Area below function in the interval. Divide [0,2] into 4 equal subintervals Left Rectangles.

[5-4] Riemann Sums and the Definition of Definite Integral Yiwei Gong Cathy Shin.

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.

Finite Sums, Limits, and Definite Integrals.  html html.

Application of the Integral

MTH1170 Integrals as Area.

Do Now - #22 and 24 on p.275 Graph the function over the interval. Then (a) integrate the function over the interval and (b) find the area of the region.

Riemann Sums and the Definite Integral

Midpoint and Trapezoidal Rules

A Preview of Calculus Lesson 1.1.

Section 6. 3 Area and the Definite Integral Section 6

Riemann Sums Approximate area using rectangles

Approximating Definite Integrals. Left Hand Riemann Sums.

Approximating Definite Integrals. Left Hand Riemann Sums.

Let V be the volume of the solid that lies under the graph of {image} and above the rectangle given by {image} We use the lines x = 6 and y = 8 to divide.

Section 3.2 – Calculating Areas; Riemann Sums

Area and the Definite Integral

Find an approximation to {image} Use a double Riemann sum with m = n = 2 and the sample point in the lower left corner to approximate the double integral,

Limits of Riemann’s Sum

Accumulation AP Calculus AB Days 11-12

Chapter 3: Calculus~Hughes-Hallett

F’ means derivative or in this case slope.

7.2 Definite Integration.

Applications of Integration

4.3 Day 1 – Day 1 Review Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening and is recorded in the table. Estimate the.

4.3 Day 1 Exit Ticket for Feedback

Section 3.2 – Calculating Areas; Riemann Sums

Arc Length … x y a b xi ... Pi P0 P1 Pn

76 – Riemann Sums – Rectangles Day 2 – Tables Calculator Required

AP Calculus December 1, 2016 Mrs. Agnew

Area Under a Curve Riemann Sums.

Chapter 5 Integration.

6-2 definite integrals.

Jim Wang Mr. Brose Period 6

Presentation transcript:

Lesson 16 and 17 Area and Riemann Sums MATH 1314 Lesson 16 and 17 Area and Riemann Sums

Approximating Area

RectangleSum(f(x), 0, 2, 4, 0)

RectangleSum(f(x), -2.82, 1.33, 50, .5)

RectangleSum(f(x), 0.075, 8.21, 12, 0)

Upper and Lower Sums GeoGebra

UpperSum(f(x), -3, 5, 35) LowerSum(f(x), -3, 5, 35)

Popper 20: How would read the following integral? −2 5 (𝑥+1) 2 𝑑𝑥 The slope of the tangent line between x= -2 and x = 5 The rate of increase between x = -2 and x = 5 The area between the graph and the x-axis from x = -2 to x = 5 The arc length of the graph between x = -2 and x = 5 2. Calculate the approximate area using 5 left corner rectangles. 3. Calculate the approximate area using 5 right corner rectangles. 4. Calculate the upper sum of 5 rectangles. 5. Calculate the lower sum of 5 rectangles. a. 72.33 b. 99.12 c. 100.30 d. 50.12 e. 48.72

Popper 20….last question: 6. How you would change this method to be calculate an exact area? Use midpoint rectangles. Average all the approximations together Double the number of rectangles. Increase the number of rectangles to be a limit at infinity.