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Chapter 6 | Applications of Integration
⎝ x – , y –
centroid ⎛
⎞ ⎠ for the given shapes. Use symmetry to
help locate the center of mass whenever possible.
285. [T] Quarter-circle: y = 1− x 2 , y =0, and x =0 286. [T] Triangle: y = x , y =2− x , and y =0 287. [T] Lens: y = x 2 and y = x 288. [T] Ring: y 2 + x 2 =1 and y 2 + x 2 =4 289. [T] Half-ring: y 2 + x 2 =1, y 2 + x 2 =4, and y =0 290. Find the generalized center of mass in the sliver between y = x a and y = x b with a > b . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis. 291. Find the generalized center of mass between y = a 2 − x 2 , x =0, and y =0. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis. 292. Find the generalized center of mass between y = b sin( ax ), x =0, and x = π a . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y -axis. 293. Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a is positioned with the left end of the circle at x = b , b >0, and is rotated around the y -axis.
⎝ x – , y –
294. Find the center of mass ⎛ ⎞ ⎠ for a thin wire along the semicircle y = 1− x 2 with unit mass. ( Hint: Use the theorem of Pappus.)
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