Slide 1 / 85 Slide 2 / 85 Table of Contents Riemann Sums Trapezoid Rule Accumulation Function Antiderivatives & Definite Integrals Mean Value Theorem & Average Value Definite Integrals Fundamental Theorem of Calculus Slide 3 / 85 Slide 4 / 85 Consider the following velocity graph: 30 mph Riemann Sums 5 hrs How far did the person drive? The area under the velocity graph is the total Return to distance traveled. Integration is used to find Table of the area. Contents Slide 5 / 85 Slide 6 / 85 But we seldom travel at a constant speed. George Riemann (Re-mon) studied making these curves into a series of rectangles. 50 mph 5 hrs So the area under the curve would be The area under this graph is still the distance traveled the sum of areas of the rectangles but we need more than multiplication to find it. this is called Riemann Sums.
Slide 7 / 85 Slide 8 / 85 Riemann Sums Example: Find the area y = x 2 and the x-axis [0,1] using Riemann Sums, or Rectangular Approximation Method, is Riemann Sums and 4 partitions. calculated by drawing rectangles from the x-axis up to the curve. The question is: What part of the "top" of the rectangle should lie on the curve? Found the width of the rectangle: (b-a)/n= (1-0)/4 0 1/4 1/2 3/4 1 LRAM .25f(0) +.25f(.25) +.25f(.5)+.25f(.75) .25(0) + .25(.0625) + .25(.25) + .25(.5625) The right The left hand The middle. 0 + .015625 + .0625 + .140625 hand corner. corner. (MRAM) .21875 ≈ .219 (RRAM) (LRAM) Is this approximation an overestimate or an underestimate? Slide 9 / 85 Slide 10 / 85 Example: Find the area y = x 2 and the x-axis [0,1] using Example: Find the area y = x 2 and the x-axis [0,1] using Riemann Sums and 4 partitions. Riemann Sums and 4 partitions. MRAM RRAM 0 1/4 1/2 3/4 1 0 1/4 1/2 3/4 1 .25f(.25) +.25f(.5)+.25f(.75) + .25f(1) .25f(1/8) + .25f(3/8) + .25f(5/8) + .25f(7/8) .25(.0625) + .25(.25) + .25(.5625) + .25(1) .25(.015625) + .25(.140625) +.25(.390625) +.25(.765625) .015625 + .0625 + .140625 + .25 .328125 ≈ .328 .46875 ≈ .469 Is this approximation an overestimate or an underestimate? This value falls between LRAM and RRAM. Slide 11 / 85 Slide 12 / 85 Q: What units should be used? A: Since the area is found by multiplying base times height, *NOTE: the units of the area are the units of the x-axis times the MRAM ≠ LRAM + RRAM units of the y-axis. 2 In our example at the beginning of the unit we had a velocity (mph) vs. time (hours) the units would then be There is a downloadable program for finding LRAM, RRAM, & MRAM.
Slide 13 / 85 Slide 14 / 85 1 When finding the area between y=3x+2 and the x-axis 2 Find the area between y=3x+2 and the x-axis [1,3] [1,3] using four partitions, how wide should each using four partitions and LRAM. interval be? Slide 15 / 85 Slide 16 / 85 3 Find the area between y=3x+2 and the x-axis [1,3] 4 When finding the area between y=3x+2 and the x-axis using four partitions and RRAM. [1,3] using four partitions and MRAM, what in the third rectangle what x should be used in f(x)? Slide 17 / 85 Slide 18 / 85 5 Find the area between y=3x+2 and the x-axis [1,3] We can write the four areas using using four partitions and MRAM. where a k is the area of the k th rectangle. It is just another way of writing what we just did. Σ is the Greek letter sigma and stands for the summation of all the terms evaluated at starting with the bottom number and going through to the top.
Slide 19 / 85 Slide 20 / 85 Equivalent Formulas Selected Rules for Sigma 1 st n integers: 1 st n squares: 1 st n cubes: Slide 21 / 85 Slide 22 / 85 6 7 55 -91 Slide 23 / 85 Slide 24 / 85 8 9 1million -20
Slide 25 / 85 Slide 26 / 85 Example: Find the area y = x 2 and the x-axis [0,1] using 4 partitions. Trapezoid Rule 0 1/4 1/2 3/4 1 Why were areas found using RAM only estimates? Return to Table of How could we draw lines to improve our estimates? Contents What shape do you get? Slide 27 / 85 Slide 28 / 85 Example: Find the area y = x 2 and the x-axis [0,1] using 4 partitions and the trapezoids. *NOTE: Trapezoid Approximation = LRAM + RRAM 2 Trapezoids 0 1/4 1/2 3/4 1 We could make our approximation even closer if we used parabolas instead of lines as the tops of our intervals. This is called Simpson's Rule but this is not on the AP Calc AB exam. Slide 29 / 85 Slide 30 / 85 10 The area between y = and the x-axis [1,3] is 11 The area between y = and the x-axis [1,3] is approximated with 5 partitions and trapezoids. What is approximated with 5 partitions and trapezoids. What is the area of the 5 th trapezoid? the height of each trapezoid?
Slide 31 / 85 Slide 32 / 85 13 What is the approximate area using the trapezoids that 12 The area between y = and the x-axis [1,3] is are drawn? approximated with 5 partitions and trapezoids. What is the approximate area? Slide 33 / 85 Slide 34 / 85 14 What is the approximate fuel consumed using the trapezoids rule for this hour flight? In the last 2 responder questions, the partitioned intervals weren't uniform. Time (minutes) Rate of Consumption (gal/min) 0 0 The AP will use both. So don't assume. 10 20 25 30 40 40 60 45 Slide 35 / 85 Slide 36 / 85 So far we have been summing areas using Σ . Gottfried Leibniz, a German mathematician, came up with a symbol you're going to see a lot of: ∫ . Accumulation It is actually the German S instead of the Greek. It still means summation. Function As a point of interest, we use the German notation in calculus because Leibniz was the first to publish. Sir Isaac Newton is now given the credit for unifying calculus because his notes predate Return to Leibniz's. Table of Contents
Slide 37 / 85 Slide 38 / 85 Another way we can calculate area under a function is to use geometry. V (m/s) The symbol notation for the area from zero to 3: What's happening during t=0 to t=3? What is the area of t=0 to t=3? What does the area mean? "The area from t=0 to t=3 is the integral from 0 to 3 of the What is the acceleration at t=3? t (sec) velocity function with respect to t." What is happening during t=3 to t=6? In general: What is the area of t=3 to t=6? What does this area mean? Where is the object at t=6 in relation to where it was at t=0? Slide 39 / 85 Slide 40 / 85 When solving an accumulation function: (direction)(relation to x-axis)(area) What is the area from x=-4 to x=0? What does this area mean? What is the object's position at x=0 in relation to x=-4? What is the object's position at x=-4 in relation to x=0? Slide 41 / 85 Slide 42 / 85 15 16 semicircle semicircle
Slide 43 / 85 Slide 44 / 85 17 18 semicircle semicircle Slide 45 / 85 Slide 46 / 85 19 Antiderivatives semicircle & Definite Integrals Return to Table of Contents Slide 47 / 85 Slide 48 / 85 Properties of Definite Integrals Area under the curve of f(x) from a to b is We have been using geometry to find A. The antiderivative of f(x) can also be used.
Slide 49 / 85 Slide 50 / 85 20 21 Slide 51 / 85 Slide 52 / 85 22 23 Slide 53 / 85 Slide 54 / 85 Antiderivative Rules Where F(x) is the anti derivative of f(x).
Slide 55 / 85 Slide 56 / 85 Examples: 24 Notice how C always disappears? We don't need when we do definite integrals. Slide 57 / 85 Slide 58 / 85 25 26 Slide 59 / 85 Slide 60 / 85 27 28
Slide 61 / 85 Slide 62 / 85 29 30 Slide 63 / 85 Slide 64 / 85 *Spoiler Alert* *Spoiler Alert* *Spoiler Alert* *Spoiler Alert* The graphing calculator also has a built-in integration function. You can do definite integrals on your graphing calculator. MATH -> 9:fnInt( For the TI-84: use the MATH key 9: fnInt( depending on the version of the operating system: For example: fnInt( or For example, integrate x 2 - 3 from 1 to 4 with respect to x. Depending on which version of the operating system you have: fnInt(x 2 - 3,x,1,4) Slide 65 / 85 Slide 66 / 85 Think back to this example from the beginning of the unit. 50 mph Mean Value Theorem & 5 hrs Average Value The area under this graph is the distance traveled. What is the average rate of speed for this trip? Total distance / total time. How can we use integrals to express this? Return to Table of Contents
Slide 67 / 85 Slide 68 / 85 Average Value How can this be used? Think Easy Pass on the Parkway or Turnpike. Your time through a booth is recorded and recorded again at the next booth which is a known distance apart. Distance b-a oops, ticket in the mail! Slide 69 / 85 Slide 70 / 85 Using that same graph of our trip, let' The Mean Value Theorem say the average speed was 50 mph. 50 mph 5 hrs or This means there are some times where are speed is less than 50 mph and some where it is more than 50 mph. By the Intermediate Value Theorem, There must be at least one time, c, when are speed is exactly 50 mph. This is the Mean Value Theorem Slide 71 / 85 Slide 72 / 85 Ex: Find c that satisfies the Mean Value Theorem for Ex: Find c that satisfies the Mean Value Theorem for since -2 is not on [-1,2] c=2/3
Slide 73 / 85 Slide 74 / 85 Ex: Find c that satisfies the Mean Value Theorem for 31 Find c that satisfies the Mean Value Theorem for Slide 75 / 85 Slide 76 / 85 32 Find c that satisfies the Mean Value Theorem for 33 Find c that satisfies the Mean Value Theorem for Slide 77 / 85 Slide 78 / 85 34 Find c that satisfies the Mean Value Theorem for Fundamental Theorem of Calculus Return to Table of Contents
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