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7.1- Nicole 7.2- Valerie, Maryalice 7.3- Nathaniel (ME!) 7.4- Yasmeen Nicole Stevenson 7.1 Review #1: Question: Answer: #2 Question: Answer: #3 Question: Answer: 7.2 Review Valerie 1. Question: Find the area of the shaded region analytically: (refer to page 395 #5 for graph) Solution: 𝐵 ∫𝐴 𝑓(𝑥) − 𝑔(𝑥)𝑑𝑥 4 3 𝑥 3 1 − 5 𝑥5 4 2 2 ∫−2 2𝑥 2 𝑑𝑥 − ∫−2 𝑥 4 − 2𝑥 2 𝑑𝑥 1 4 ∫ 2𝑥 2 − 𝑥 4 + 2𝑥 2 𝑑𝑥 1 [3 (2)3 − 5 (2)5 ] − [3 (−2)3 − 5 (−2)5 ] = ∫ 4𝑥 2 − 𝑥 4 128 15 2. Question: Find the area of the regions enclosed by the lines and curves: 𝑦 = 𝑥 2 − 2 𝑎𝑛𝑑 𝑦 = 2 Solution: 𝑥2 − 2 = 2 𝑥2 − 4 = 0 2 𝑥 = 2, −2 First; Find the x-intercepts 2 ∫ 2 − 𝑥 2 + 2𝑑𝑥 −2 ∫ 4 − 𝑥 2 𝑑𝑥 = 4𝑥 − −2 𝑥3 3 [4(2) − (2)3 (−2)3 32 ] − [4(−2) − ]= 3 3 3 3. Question: Find the area of the regions enclosed by the lines and curves: 𝑥 + 𝑦 2 = 0 𝑎𝑛𝑑 𝑥 + 3𝑦 2 = 2 Solution: 𝑥 = −𝑦 2 𝑥 = 2 − 3𝑦 2 First; Solve equations for x −𝑦 2 = 2 − 3𝑦 2 1 𝑦 = 1, −1 Next; Set both equal and solve for y ∫−1[(2 − 3𝑦 2 ) − (−𝑦 2 )]𝑑𝑦 1 2 ∫−1 2 − 2𝑦 2 𝑑𝑦 = 2𝑦 − 3 𝑦 3 2 2 8 [2(1) − 3 (1)3 ] − [2(−1) − 3 (−1)3 ] = 3 7.2 Two Maryalice Weed AP Calculus Period 6 Section 7.2 Review 1. Find the area of the region R in the first quadrant that is bounded above by 𝑦 = √𝑥 and below by the x-axis and the line 𝑦 = 𝑥 − 2 Solution: 10/3 2. Find the area between the x-axis and the function 𝑦 = √9 − 𝑥 2 over the interval [-3, 3] Solution: 5/6 or 9𝜋/2 3. Find the area of the region first quadrant bounded by the x-axis and the graphs of √𝑥 − 2 and 𝑦 = 𝑥 − 10 by subtracting the area of the triangular region from the area under the square root curve. Solution: 19.88 7.3 Nathaniel Kreiman Question: Find the volume of the solid that lies between the planes perpendicular to the x-axis at x=0 and x=4. The cross sections perpendicular to the axis on the interval [0,4] are squares whose diagonals run from 𝑦 = −√𝑥 to 𝑦 = √𝑥 Solution: Question: Find the volume of the solid created by rotating the given region about the x-axis. The region is bounded by 𝑥 + 2𝑦 = 2, the x-axis and the y-axis. Solution: Question: Find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis. 𝑦 = 𝑥 2 + 1, 𝑦 = 𝑥 + 3. Solution: (Following two pages.) 7.4 Question: Solution: Question: Solution: Question: Solution: