Chapter 6 | Applications of Integration
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6.6 EXERCISES For the following exercises, calculate the center of mass for the collection of masses given. 254. m 1 =2 at x 1 =1 and m 2 =4 at x 2 =2 255. m 1 =1 at x 1 =−1 and m 2 =3 at x 2 =2 256. m =3 at x =0, 1, 2, 6 257. Unit masses at ( x , y ) = (1, 0), (0, 1), (1, 1) 258. m 1 =1 at (1, 0) and m 2 =4 at (0, 1) 259. m 1 =1 at (1, 0) and m 2 =3 at (2, 2) For the following exercises, compute the center of mass x – . 260. ρ =1 for x ∈ (−1, 3) 261. ρ = x 2 for x ∈ (0, L ) 262. ρ =1 for x ∈ (0, 1) and ρ =2 for x ∈ (1, 2) 263. ρ = sin x for x ∈ (0, π )
For the following exercises, use a calculator to draw the region, then compute the center of mass ⎛ ⎝ x – , y – ⎞ ⎠ . Use symmetry to help locate the center of mass whenever possible. 273. [T] The region bounded by y =cos(2 x ), x = − π 4 , and x = π 4 274. [T] The region between y =2 x 2 , y =0, x =0, and x =1
x 2 and y =5
275. [T] The region between y = 5 4
276. [T] Region between y = x , y = ln( x ), x =1, and x =4 277. [T] The region bounded by y =0, x 2 4 + y 2 9 =1 278. [T] The region bounded by y =0, x =0, and x 2 4 + y 2 9 =1 279. [T] The region bounded by y = x 2 and y = x 4 in the first quadrant For the following exercises, use the theorem of Pappus to determine the volume of the shape. 280. Rotating y = mx around the x -axis between x =0 and x =1 281. Rotating y = mx around the y -axis between x =0 and x =1 282. A general cone created by rotating a triangle with vertices (0, 0), ( a , 0), and (0, b ) around the y -axis. Does your answer agree with the volume of a cone? 283. A general cylinder created by rotating a rectangle with vertices (0, 0), ( a , 0), (0, b ), and ( a , b ) around the y -axis. Does your answer agree with the volume of a cylinder? 284. A sphere created by rotating a semicircle with radius a around the y -axis. Does your answer agree with the volume of a sphere? For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area M and the
264. ρ =cos x for x ∈ ⎛
⎞ ⎠
⎝ 0, π 2
265. ρ = e x for x ∈ (0, 2) 266. ρ = x 3 + xe − x for x ∈ (0, 1) 267. ρ = x sin x for x ∈ (0, π )
268. ρ = x for x ∈ (1, 4) 269. ρ = ln x for x ∈ (1, e )
For the following exercises, compute the center of mass ⎛ ⎝ x – , y – ⎞ ⎠ . Use symmetry to help locate the center of mass whenever possible. 270. ρ =7 in the square 0≤ x ≤1, 0≤ y ≤1 271. ρ =3 in the triangle with vertices (0, 0), ( a , 0), and (0, b ) 272. ρ =2 for the region bounded by y =cos( x ), y =−cos( x ), x = − π 2 , and x = π 2
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