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Section 8.5 Riemann Sums and the Definite Integral
represents the area between the curve 3/x and the x-axis from x = 4 to x = 8
Four Ways to Approximate the Area Under a Curve With Riemann Sums Left Hand Sum Right Hand Sum Midpoint Sum Trapezoidal Rule
Approximate using left-hand sums of four rectangles of equal width 1.Enter equation into y1 2.2 nd Window (Tblset) 3.Tblstart: 4 4.Tbl: nd Graph (Table)
Approximate using right-hand sums of four rectangles of equal width 1.Enter equation into y1 2.2 nd Window (Tblset) 3.Tblstart: 5 4.Tbl: nd Graph (Table)
Approximate using midpoint sums of four rectangles of equal width 1.Enter equation into y1 2.2 nd Window (Tblset) 3.Tblstart: Tbl: nd Graph (Table)
Approximate using trapezoidal rule with four equal subintervals 1.Enter equation into y1 2.2 nd Window (Tblset) 3.Tblstart: 4 4.Tbl: nd Graph (Table)
Approximate using left-hand sums of four rectangles of equal width
Approximate using trapezoidal rule with n = 4
For the function g(x), g(0) = 3, g(1) = 4, g(2) = 1, g(3) = 8, g(4) = 5, g(5) = 7, g(6) = 2, g(7) = 4. Use the trapezoidal rule with n = 3 to estimate
If the velocity of a car is estimated at estimate the total distance traveled by the car from t = 4 to t = 10 using the midpoint sum with four rectangles
Riemann Sums and the Definite Integral. represents the area between the curve 3/x and the x-axis from x = 4 to x = 8.
Section 7.6 – Numerical Integration. represents the area between the curve 3/x and the x-axis from x = 4 to x = 8.
Section 3.2 – Calculating Areas; Riemann Sums Pick up notes out of your folder.
Section 7.6 – Numerical Integration Day 5: I can integrate definite integrals using Left Hand Sum, Right Hand Sum, Midpoint Sum, and Trapezoidal Rule.
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.
5.1 Estimating with Finite Sums Greenfield Village, Michigan.
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.
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.
5.1 Estimating with Finite Sums Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 Greenfield Village, Michigan.
Finding the area under a curve: Riemann, Trapezoidal, and Simpsons Rule Adguary Calwile Laura Rogers Autrey~ 2 nd Per. 3/14/11.
RIEMANN SUMS AP CALCULUS MS. BATTAGLIA. Find the area under the curve from x = 0 to x = 35. The graph of g consists of two straight lines and a semicircle.
Slide 5- 1 What you’ll learn about Distance Traveled Rectangular Approximation Method (RAM) Volume of a Sphere Cardiac Output … and why Learning about.
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.
The Definite Integral. Area below function in the interval. Divide [0,2] into 4 equal subintervals Left Rectangles.
Section 4.3 Day 2 Riemann Sums & Definite Integrals AP Calculus BC.
7.2: Riemann Sums: Left & Right-hand Sums. Today you’ll learn how to estimate the integral the same way Riemann approached it. Left-Hand Right-Hand.
Definite Integral df. f continuous function on [a,b]. Divide [a,b] into n equal subintervals of width Let be a sample point. Then the definite integral.
5.2 – The Definite Integral. Introduction Recall from the last section: Compute an area Try to find the distance traveled by an object.
THE DEFINITE INTEGRAL RECTANGULAR APPROXIMATION, RIEMANN SUM, AND INTEGRTION RULES.
5.1 Estimating with Finite Sums Objectives SWBAT: 1) approximate the area under the graph of a nonnegative continuous function by using rectangular approximation.
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.
2/28/2016 Perkins AP Calculus AB Day 15 Section 4.6.
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,
5.1 Areas and Distances 1 Dr. Erickson. 5.1 Areas and Distances2 time velocity After 4 seconds, the object has gone 12 feet. Since rate. time = distance:
SECTION 5.1: ESTIMATING WITH FINITE SUMS Objectives: Students will be able to… Find distance traveled Estimate using Rectangular Approximation Method Estimate.
Numerical Integration. Integration/Integration Techniques/Numerical Integration by M. Seppälä Definite Integrals.
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Definite Integrals & Riemann Sums. Definition of a Definite Integral A definite integral of a function y = f(x) on an interval [a, b] is the signed (
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?
Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?
Warm Up – Calculator Active 1) A particle moves along the x-axis so that at any time t > 0, its velocity is given by v(t) = cos (0.9t). What is.
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A REA A PPROXIMATION 4-E Riemann Sums. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.
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Linear Approximation It is a way of finding the equation of a line tangent to a curve and using it to approximate a y value of the curve.
When you see… Find the zeros You think…. To find the zeros...
5.6 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington.
5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington.
Trapezoidal Approximation Objective: To relate the Riemann Sum approximation with rectangles to a Riemann Sum with trapezoids.
Review Distance Traveled Area Under a curve Antiderivatives Calculus.
Riemann Sums Approximating Area. One of the classical ways of thinking of an area under a curve is to graph the function and then approximate the area.
1 5.2 – The Definite Integral. 2 Review Evaluate.
1 Copyright © 2015, 2011 Pearson Education, Inc. Chapter 5 Integration.
Solving Equations by Graphing - Intersection Method Let Y1 = left hand side of the equation. Let Y2 = right hand side of the equation. Example Solve 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.
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