Chapter 6 | Applications of Integration
683
189. [T] x =4 y from y =0 to y =2
207. The base of a lamp is constructed by revolving a quarter circle y = 2 x − x 2 around the y -axis from x =1 to x =2, as seen here. Create an integral for the surface area of this curve and compute it.
190. [T] x = ln( y ) on y = 1 e to y = e For the following exercises, find the surface area of the volume generated when the following curves revolve around the x -axis. If you cannot evaluate the integral exactly, use your calculator to approximate it.
191. y = x from x =2 to x =6 192. y = x 3 from x =0 to x =1 193. y =7 x from x =−1 to x =1
208. A light bulb is a sphere with radius 1/2 in. with the bottom sliced off to fit exactly onto a cylinder of radius 1/4 in. and length 1/3 in., as seen here. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is 1/4 in. Find the surface area (not including the top or bottom of the cylinder).
194. [T] y = 1 x 2
from x =1 to x =3
195. y = 4− x 2 from x =0 to x =2 196. y = 4− x 2 from x =−1 to x =1
197. y =5 x from x =1 to x =5 198. [T] y = tan x from x = − π 4 to
x = π 4
For the following exercises, find the surface area of the volume generated when the following curves revolve around the y -axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. 199. y = x 2 from x =0 to x =2
209. [T] A lampshade is constructed by rotating y =1/ x around the x -axis from y =1 to y =2, as seen here. Determine how much material you would need to construct this lampshade—that is, the surface area—accurate to four decimal places.
x 2 + 1 2
200. y = 1 2
from x =0 to x =1
201. y = x +1 from x =0 to x =3
202. [T] y = 1 x from x = 1 2
to x =1
203. y = x 3 from x =1 to x =27 204. [T] y =3 x 4 from x =0 to x =1 205. [T] y = 1 x from x =1 to x =3 206. [T] y =cos x from x =0 to x = π 2
210. [T] An anchor drags behind a boat according to the function y =24 e − x /2 −24, where y represents the depth beneath the boat and x is the horizontal distance of the anchor from the back of the boat. If the anchor is 23 ft below the boat, how much rope do you have to pull to reach the anchor? Round your answer to three decimal places.
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