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Unit: Integration MATH 3 Scott Pauls Department of Mathematics Dartmouth College Instructor’s overview - 1 These slides are not meant to be prescriptive, but serve as a skeletal outline of the course syllabus using active learning methods throughout the class. In these slides we assume that most classes have the same general format: 1. (5-10 minutes) A brief introductory mini-lecture, often recapitulating and extending the end of the previous class, leading into the first group activity. 2. (10-15 minutes) Group activity meant to explore a new topic or extend and old one. 3. (5-10 minutes) A short follow up on the results and lead-in to a second piece which is sometimes lecture/discussion and sometimes another period for group work. 4. (10-15 minutes) Whatever the second piece is. 5. (5-10 minutes) Follow up on second piece 6. (5-10 minutes) A short introduction to the homework, preparation, and next piece of material for the next class. Instructor’s overview – 2 The problems used in the slides below are indicative what level we expect throughout the course. Instructors will want to supplement these with both easier and harder examples as class progress dictates. One goal in providing these templates is to help ensure a uniform level of instruction to which we can test. For each set of group work, there are a sequence of questions, ordered by difficulty. Each one is aimed to build up to the next with the goal of completing the last one, which is the most abstract. Those that are circled are candidates for write-up and grading by group. Average final mastery by topic (F15) Midterm 1 Midterm 2 Anti-differentiation and indefinite integrals 𝑑 𝐹 𝑑𝑥 𝑥 = 𝐹′ 𝑥 = 𝑓 𝑥 ∫ 𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶 Examples: Function Antiderivative sin(𝑥) 𝑥2 +𝐶 2 − cos 𝑥 + 𝐶 cos(𝑥) sin 𝑥 + 𝐶 𝑒𝑥 𝑒𝑥 + 𝐶 𝑥 Rectilinear motion If 𝑝(𝑡) gives the position of an object in motion at time 𝑡, then ′ we know that 𝑣 𝑡 = 𝑝′(𝑡) is its velocity and 𝑎 𝑡 = 𝑣 𝑡 = 𝑝′′(𝑡) is its acceleration. Using anti-differentiation, we can start with the acceleration due to gravity and work our way backwards. If 𝑎 𝑡 = −9.8 𝑚/𝑠 2 then 𝑣 𝑡 = −9.8𝑡 + 𝐶. What is the constant? If we know 𝑣 0 = 𝑣0 then 𝐶 = 𝑣0 . Continuing, we anti-differentiate again yielding 9.8 2 𝑝 𝑡 =− − 𝑡 + 𝑣0 𝑡 + 𝐷. 2 Again, if we know 𝑝 0 = 𝑝0 then 𝐷 = 𝑝0 . 𝑣0 𝑝0 Group Work: The Siege of Syracuse 1. If the trajectory of a boulder is given by (𝑥′′ 𝑡 , 𝑦 𝑡 ) with 𝑥2 0 = 𝑦 0 = 0 and we set 𝑦 0 = −9.8 𝑚/𝑠 and 𝑥 0 = 0, what are the functions 𝑥(𝑡) and 𝑦 𝑡 ? 2. If, as in the diagram, the catapult launches with a fixed speed but variable angle 𝜃, what ′ are 𝑥′(0) and 𝑦 0 ? 𝑠 3. 4. Given all of this information, where does the boulder hit the x-axis? 𝜃 𝑥′ 𝑡 Suppose a ship sits at (100 𝑚, 0) and you have a catapult that launches with an initial speed of 𝑠 = 25. How do we set up the catapult to hit the ship? (𝑥 ′ 𝑡 , 𝑦 ′ 𝑡 ) 𝑦′ 𝑡 Challenge problems 1. In the catapult problem, we fixed the trajectory on a plane. Can you generalize to all of three-dimensional space? 2. What is the maximum range of a catapult with speed 𝑠? How is that range achieved? 3. What are the benefits and drawbacks of placing catapult higher or lower than the sea level? Finding areas What is the area of an object in the plane? Simplification: What is the area between a curve 𝑦 = 𝑓(𝑥) and the 𝑥-axis? A question we can answer: What is the area of a rectangle? Riemann sums 𝑦 = 𝑒 −𝑥 𝐴 = ℎ𝑤 = 𝑒 −2 ⋅ 1 With four rectangles (as in the figure) the left endpoint Riemann sum is 𝑒 −0 ⋅ 1 + 𝑒 −1 ⋅ 1 + 𝑒 −2 ⋅ 2 + 𝑒 −3 ⋅ 1 ℎ =𝑓 2 = 𝑒 −2 4 𝑒− = 𝑖=1 𝑖−1 Δ𝑥 w = Δ𝑥 = 1 Riemann sums For 𝑛 rectangles, Δ𝑥 = 𝑏−𝑎 𝑛 . Sample points are given by: {𝑎, 𝑎 + Δ𝑥, 𝑎 + 2Δ𝑥, … , 𝑎 + 𝑛Δ𝑥 = 𝑏} Left endpoints: 𝑎, 𝑎 + Δ𝑥, 𝑎 + 2Δ𝑥, … , 𝑎 + 𝑛 − 1 Δ𝑥 Right endpoints: {𝑎 + Δ𝑥, 𝑎 + 2Δ𝑥, … , 𝑎 + 𝑛Δ𝑥 = 𝑏} Area of 𝒌𝒕𝒉 box: 𝑓 Left endpoints: 𝑓 𝑎 + 𝑘 − 1 Δx Δ𝑥 = Right endpoints: 𝑓 𝑎 + kΔx Δ𝑥 = 𝑓 𝑎+ 𝑘 𝑏−𝑎 𝑎+ 𝑛 𝑛 Riemann sums: Left endpoints: Right endpoints: 𝑛 𝑘=1 𝑓 𝑘−1 𝑏−𝑎 𝑏−𝑎 𝑛 𝑛 𝑘 𝑏−𝑎 𝑏−𝑎 + 𝑛 𝑛 𝑎+ 𝑛 𝑘=1 𝑓 𝑎 𝑘−1 𝑏−𝑎 𝑛 𝑛 𝑏−𝑎 𝑏−𝑎 Riemann sums Group work Let 𝑓 𝑥 = 𝑥 2 − 𝑥 + 1. 3 1. Estimate ∫1 𝑓 𝑥 𝑑𝑥 using Riemann sums with right endpoints with 3 rectangles. 2. Write down the Riemann sum with left endpoints for ∫1 𝑓 𝑥 𝑑𝑥 with 𝑛 rectangles. 3. Write down the Riemann sum with right endpoints for ∫𝑎 𝑓 𝑥 𝑑𝑥 with 6 rectangles. 4. Write down the Riemann sum with left endpoints for ∫𝑎 𝑓 𝑥 𝑑𝑥 with 𝑛 rectangles. 3 𝑏 𝑏 Evaluating definite integrals Find I = 4 ∫1 𝑥 3 − 4𝑥 𝑑𝑥. Left hand endpoints: 𝐼 = lim 𝑛→∞ 𝑛 𝑖=1 1 + (𝑖 − 3 3 1) 𝑛 − 4 1 + (𝑖 − Right hand endpoints: 𝐼 = lim 𝑛→∞ 𝑛 𝑖=1 1+ 3 3 𝑖 𝑛 −4 1+ 3 𝑖 𝑛 3 𝑛 3 1) 𝑛 3 𝑛 Definite integrals Group work We’ll assign each group one of three simple functions: 𝑓 𝑥 = 𝑥, 𝑥 2 , or 𝑥 3 . For your function, complete the following problems: 𝑏 1. Write down the definition of ∫𝑎 𝑓 𝑥 𝑑𝑥 using Riemann sums. 2. Find Δ𝑥 for the sum using 𝑛 rectangles and compute the heights of the rectangles using right or left-hand endpoints (your choice). 3. Simplify the summands using algebra and the formulae above. 4. Compute the resulting limits. 5. Compute ∫𝑎 𝑓 𝑥 𝑑𝑥 using Riemann sums. 𝑏 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus Group Work Pick on of the graphs to the right to 𝑥be 𝑓(𝑥) and let 𝑔 𝑥 = ∫0 𝑓 𝑡 𝑑𝑡. 1. At what values of 𝑥 do local maxima and minima of 𝑔(𝑥) occur? 2. Where does 𝑔(𝑥) obtain its global maximum on this interval? 3. On which intervals is 𝑔 𝑥 concave up and concave down? 4. Sketch a graph of 𝑔 𝑥 . Calculating areas with Riemann sums Consider the Gaussian function 2 𝑥−𝜇 1 𝑓 𝑥 = 𝑒 2𝜎2 2𝜎 2 𝜋 1. Integration techniques: 𝑢-substitution FTC applied to the chain rule: ′ 𝑓 𝑔 𝑥 = 𝑓 ′ 𝑔 𝑥 𝑔′ 𝑥 𝑏 𝑓 ′ 𝑔 𝑥 𝑔′ 𝑥 𝑑𝑥 = 𝑓 𝑔 𝑏 ⟹ 𝑎 − 𝑓(𝑔 𝑎 ) Substitution Group work Integration techniques: integration by parts FTC applied to the product rule: ′ 𝑓 𝑥 𝑔 𝑥 = 𝑓 ′ 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔′ 𝑥 𝑏 𝑓 ′ 𝑥 𝑔 𝑥 + 𝑓 𝑥 𝑔′ 𝑥 𝑑𝑥 = 𝑓 𝑏 𝑔 𝑏 − 𝑓 𝑎 𝑔(𝑎) ⟹ 𝑎 Rewritten: 𝑏 ∫𝑎 𝑓 ′ 𝑥 𝑔 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑔 𝑥 |𝑏𝑎 𝑏 ′ − ∫𝑎 𝑓 𝑥 𝑔 𝑥 𝑑𝑥 Integration by parts Group work