Slide 1 / 175 Slide 2 / 175 AP Calculus AB Integration 2015-11-24 www.njctl.org Slide 3 / 175 Slide 4 / 175 click on the topic to go Table of Contents Integration to that section Mathematicians spent a lot of time working with the topic of Riemann Sums derivatives, describing how functions change at any given instant. Trapezoid Approximation They then sought a way to describe how those changes accumulate Area Under a Curve (The Definite Integral) over time, leading them to discover the calculation for area under a Antiderivatives & The Fundamental Theorem of Calculus, part II curve. This is known as integration, the second main branch of Fundamental Theorem of Calculus, part I calculus. Average Value & Mean Value Theorem for Integrals Finally, Liebniz and Newton discovered the connection between Indefinite Integrals differentiation and integration, known as the Fundamental Theorem U-Substitution of Calculus, an incredible contribution to the understanding of mathematics. Slide 5 / 175 Slide 6 / 175 Area of Curved Shapes Formulas for the area of polygons, such as squares, rectangles, triangles and trapezoids were well known in many early civilizations. However, the problem of finding area for regions with curved Riemann Sums boundaries (circles, for example) caused difficulties for early mathematicians. The Greek mathematician Archimedes proposed to calculate the area of a circle by finding the area of a polygon inscribed in the circle with the number of sides increased indefinitely. Return to Table of Contents
Slide 7 / 175 Slide 7 (Answer) / 175 Distance Using Graphs Distance Using Graphs Consider the following velocity graph: Consider the following velocity graph: Answer v(t) (mph) v(t) (mph) You can see that this number can be obtained if we calculate the 30 mph 30 mph area under the velocity graph. So, the area of the rectangle in this case represents the total distance t (hours) t (hours) traveled. 5 hrs 5 hrs [This object is a pull tab] How far did the person drive? How far did the person drive? Slide 8 / 175 Slide 9 / 175 Georg Friedrich Riemann Non-Constant Speed Georg Friedrich Bernhard Riemann was an influential German However, objects seldom travel at a constant speed. mathematician who made lasting contributions to function analysis and found an approach for approximating the total area underneath a curve by dividing the total area into a series of rectangles. v(t) (mph) 50 mph 30 mph t (hours) 5 hrs So, the area under the curve would be the sum of areas of the rectangles. Later, we will discuss how close this approximation is, and if The area under this graph is still equal to the distance traveled but there is any possibility to calculate the exact area underneath the we need more than just simple multiplication to find it. curved boundary. Slide 10 / 175 Slide 10 (Answer) / 175 Let students discuss what the area would be if it wasn't bounded by the x- Area Under the Curve Area Under the Curve axis. Hopefully they will conclude that Teacher Notes the area would be infinite. Note: When we use the language "area under the Note: When we use the language "area under the curve" we are referring to the area between the function curve" we are referring to the area between the function and the x-axis. and the x-axis. vs. Finite Area Area [This object is a pull tab]
Slide 11 / 175 Slide 12 / 175 Riemann Sums RAM - Rectangular Approximation Method Rectangular Approximation Method is a way to estimate area by Example: Approximate the area under the curve y = x 2 on [0,1] with drawing rectangles from the x-axis up to the curve. a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). The question is: What part of the "top" of the rectangle should lie on the curve? Also, how many rectangles should be used? Is this approximation an overestimate or an underestimate? Explain. The right The left hand The middle hand corner corner (MRAM) (RRAM) (LRAM) Slide 13 / 175 Slide 14 / 175 Riemann Sums Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain. Finally, calculate the area using LRAM. Have students discuss why this is an under approximation. Slide 15 / 175 Slide 16 / 175 Riemann Sums Riemann Sums Example: Approximate the area under the curve y = x 2 on [0,1] with Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and left a Riemann sum using 4 sub-intervals (rectangles) and right endpoints (LRAM). Is this approximation an overestimate or an endpoints (RRAM). underestimate? Explain. We calculated the area using LRAM to be If we look at our graph, we can see that all of the rectangles are below Is this approximation an our curve. Therefore, this overestimate or an underestimate? approximation is an underestimate.
Slide 16 (Answer) / 175 Slide 17 / 175 Riemann Sums Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and midpoints (MRAM). Slide 17 (Answer) / 175 Slide 18 / 175 Riemann Sums *NOTE: LRAM + RRAM MRAM ≠ 2 Slide 18 (Answer) / 175 Slide 19 / 175 1 When approximating the area under the curve y=3x+2 Riemann Sums on [1,4] using four rectangles, how wide should each sub-interval be? Technically, for some functions, Teacher Notes MRAM could be the average of LRAM and RRAM; however, for most *NOTE: functions given in practice and on the exam, this will not be the case. LRAM + RRAM MRAM ≠ 2 Have a discussion with students about why this can't always be true. [This object is a pull tab]
Slide 19 (Answer) / 175 Slide 20 / 175 2 Approximate the area under y=3x+2 on [1,4] using six rectangles and LRAM. Slide 20 (Answer) / 175 Slide 21 / 175 3 Find the area under the curve on [-3,2] using five sub-intervals and RRAM. Slide 21 (Answer) / 175 Slide 22 / 175
Slide 22 (Answer) / 175 Slide 23 / 175 Slide 23 (Answer) / 175 Slide 24 / 175 Riemann Sum Notation The following notation is used when discussing Riemann sums and approximating areas. Some questions will use this notation, so it is important to be familiar with the meaning of each symbol. Using the symbols above, can you create a mathematical relationship between all 4 of them? Slide 24 (Answer) / 175 Slide 25 / 175 Riemann Sum Notation start of interval The following notation is used when discussing Riemann sums and end of interval approximating areas. Some questions will use this notation, so it is number of sub-intervals Teacher Notes important to be familiar with the meaning of each symbol. width of each interval Some students may need guidance coming up with a relationship. The most common answer students will come up with is usually because they have been calculating the width in earlier questions. [This object is a pull tab] Using the symbols above, can you create a mathematical relationship between all 4 of them?
Slide 25 (Answer) / 175 Slide 26 / 175 Slide 26 (Answer) / 175 Slide 27 / 175 Riemann Sums with Tables Sometimes, instead of being given an equation for f(x), data points from the curve will be presented in a table. Provided the necessary information is in the table, you are still able to approximate area. Slide 28 / 175 Slide 29 / 175 Riemann Sums with Tables Example: Approximate the area under the curve, f(x), on [-2,4] using right endpoints and n=3.
Slide 30 / 175 Slide 31 / 175 Riemann Sums with Tables Riemann Sums with Tables Example: Using the subintervals in the table, approximate the area under using a left hand approximation. Answer Note: When using tabular data for Riemann Sums, not all sub- intervals need to be of equal width. If the question does not specify , then you are able to use the data provided - just make sure to account for the varying width. Slide 32 / 175 Slide 33 / 175 8 Approximate the area under the function, , based on Riemann Sums with Tables the given table values. Use a right hand approximation and 4 equal sub-intervals. Answer Example: Using the subintervals in the table, approximate the area under using a right hand approximation. Answer A D G B E H C F I Slide 34 / 175 Slide 35 / 175 9 Approximate the area under the function, , based on the given table values and intervals. Use a left hand approximation. Answer
Slide 36 / 175 Slide 37 / 175 Refresher on Summations: Slide 37 (Answer) / 175 Slide 38 / 175 Remind students how to calculate the Refresher on Summations: Sigma Notation summations, before the next slide where they will write their own to represent Riemann Sums. To represent Riemann Sums using sigma notation, we need to Answer know the number of rectangles on the interval, and height of each rectangle. We will let represent each rectangle. Example: Use sigma notation to represent the area under the curve of on using 4 equal subintervals and left endpoints. [This object is a pull tab] Slide 38 (Answer) / 175 Slide 39 / 175 12 Which of the following represents the approximate area under the curve on using midpoints and 3 equal subintervals? A B C D E
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