Use the method of your choice to find the volume of the soli

Use the method of your choice to find the volume of the solid generated by rotating the region bounded by y=5, y=x+4/x abt the line x=-1

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Solution

suppose that 1. First we find x-coordinates of points of intersection of y = 2 + x - x² y = 2 - x 2 - x = 2 + x - x² x² - 2x = 0 x (x - 2) = 0 x = 0, x = 2 So we integrate from x = 0 to x = 2 On this interval 2 + x - x² > 2 - x Axis of rotation = y-axis (line x = 0) Surface area of cylindrical shell = 2p r h where r = distance from axis of rotation = x and h = height of cylinder = (2 + x - x²) - (2 - x) = 2x - x² V = 2p ?0² r h dx V = 2p ?0² x (2x - x²) dx V = 2p ?0² (2x² - x³) dx Solving, we get V = 8p/3 ========================= let 2. y = 8vx intersects x-axis (line y=0) at point (0, 0) We integrate from x = 0 (intersection) to x = 1 (given) Axis of rotation: line = -4 Surface area of cylindrical shell = 2p r h where r = distance from axis of rotation = x - (-4) = x + 4 and h = height of cylinder = y = 8vx V = 2p ?0¹ r h dx V = 2p ?0¹ (x + 4) 8vx dx V = 2p ?0¹ (8x^(3/2) + 32x^(1/2)) dx Solving, we get V = 736p/15 ========================= so 3. Points of intersection: (0,0) and (4,4) y² = 4x ----> y = 2vx Cylinder has radius = x, height = 2vx - x V = 2p ?04 x (2vx - x) dx V = 2p ?04 (2x^(3/2) - x²) dx Solving, we get: V = 128p/15 ========================= we get 4. Points of intersection (0,0) and (1,5) Since we are rotating about a horizontal axis, we will have to integrate with respect to y We integrate from y = 0 to y = 5 y = 5x² ------> x = v(y/5) y = 5x -------> x = y/5 Cylinder has radius = y, height = v(y/5) - y/5 V = 2p ?05 y (v(y/5) - y/5) dy V = 2p ?05 (1/v5 y^(3/2) - 1/5 y²) dy V = 10p/3 -------------------- hence Of course, this would have been easier to solve using washer method Integrate from x = 0 to x = 1 Outer radius: R = 5x Inner radius: r = 5x² V = p ?0¹ (R² - r²) dx V = p ?0¹ (25x² - 25x4) dx V = 10p/3