 # Part I. Riemann Sums and Area 1. Use the definition of area and Ri...

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Part I. Riemann Sums and Area 1. Use the definition of area and Riemann sums to set up and find the net area of the function 𝑓(𝑥) = √25 − 𝑥2 from 𝑥 = 0 𝑡𝑜 𝑥 = 5 using 𝑛 = 5 rectangles using a) left endpoint sums, b) right endpoint sums, and c) midpoint sums. Round final answers to four decimals (3 pts each letter part, 9 points total). 2. Set up each Riemann Sum using the given information for parts a) through e) (1 pt each, 5 pts total) Then use Desmos to estimate the area under the function 𝑔(𝑥) = 𝑥 ln(𝑥) from 𝑥 = 0 𝑡𝑜 𝑥 = 5 using midpoint estimations for the given number of rectangles. Round each answer to four decimal places (1 pt each, 5 pts total) a) n=5 rectangles b) n=10 rectangles c) n = 50 rectangles d) n=100 rectangles e)𝑙𝑖𝑚𝑖𝑡𝑎𝑠𝑛 →∞: 2 Part II. Indefinite Integration Find the general indefinite integral for problems 3 through 5. Remember to show all steps! (5 pts each). 3. ∫(𝑥6 −2𝑥5 −𝑥3 +2)𝑑𝑥 4. ∫(sin2𝑥)𝑑𝑥 7 sin𝑥 5.∫cos(ln𝑥)𝑑𝑥 𝑥 6.∫(𝑥2 +1)(𝑥3 +3𝑥)4 𝑑𝑥 3 Part III. Definite Integration. 7. Two cars with velocities 𝑣1 𝑎𝑛𝑑 𝑣2 (in meters per second) are tested on a straight track. Consider the following information: ∫5[𝑣1(𝑡) − 𝑣2(𝑡)] 𝑑𝑡 = 10 ∫10[𝑣1(𝑡) − 𝑣2(𝑡)] 𝑑𝑡 = 30 ∫30[𝑣1(𝑡) − 𝑣2(𝑡)] 𝑑𝑡 = −5 0 0 20 a) Write a verbal interpretation of each integral above (1 pt each, 3 pts total) b) Is it possible to determine the distance between the two cars when t = 5 seconds? Explain why or why not (3 pts) c) Assume both cars start at the same time and place on the track. Which car is ahead at t = 10 seconds? How far ahead is that car compared to the other car? (3 pts) d) Suppose that Car 1 has velocity 𝑣1 and is ahead of Car 2 by 13 meters when t = 20 seconds. How far ahead or behind is Car 1 when t = 30 seconds? (3 pts) 4 For problems 8 & 9: Set up and show all calculus and algebra steps to evaluate each definite integral. Give exact answers where possible. (7 pts each) 2𝜋 3 8.∫𝜋 sec2(𝑥)𝑑𝑥 2 2 9.∫5 𝑥 𝑑𝑥 1 √2𝑥−1 5 10. Let 𝑓(𝑥) = √𝑥 + 3 𝑎𝑛𝑑 𝑔(𝑥) = 1 𝑥 + 3 For each pair of equations: a) sketch a graph of the curves defined by f(x) 2 and g(x) and the region formed by their enclosure. b) Set up and solve for the boundary points c) Determine the total area contained between the curves as determined by their boundary points (this may require multiple integral setups). (10 pts total) 6 Part IV. Extra Credit problems Partial credit may be awarded for substantial correct progress towards a correct solution. Must show all steps to receive any credit! 1. Find the value of c such that the area of the region bounded by the parabolas 𝑦 = 𝑥2 − 𝑐2 and 𝑦 = 𝑐2 − 𝑥2 is exactly 72 units. (4 pts – Challenging but doable) 2. Give two reasons why the definite integral: ∫1 1 𝑑𝑥 = −2 is incorrect. (3 pts) −1 𝑥2 7

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