Use the method of cylindrical shells to find the volume of the solid obtained by

Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. xy = 7, x = 0, y = 7, y = 9

Explanation / Answer

Use the method of cylindrical shells to find the volume V? having problems with math problems, much help appreciated 1. Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis. y=2+x-x^2, x+y=2 2. Consider the given curves to do the following y=8 sqrt x, y=0, x=1 Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = -4 3. Consider the solid obtained by rotating the region bounded by the given curves about the y-axis y^2=4x, x=y Find the volume V of the solid. 4. Consider the solid obtained by rotating the region bounded by the given curves about the x-axis. y=5x^2, y=5x, x > or = to 0 Find volume V a step by step would be nice, thanks in advance. 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 ========================= 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 ========================= 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 ========================= 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 -------------------- 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

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