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Chapter 6 | Applications of Integration
211. [T] You are building a bridge that will span 10 ft. You intend to add decorative rope in the shape of y =5 | sin ⎛ ⎝ ( xπ )/5 ⎞ ⎠ | , where x is the distance in feet from one end of the bridge. Find out how much rope you need to buy, rounded to the nearest foot. For the following exercises, find the exact arc length for the following problems over the given interval. 212. y = ln(sin x ) from x = π /4 to x = (3 π )/4. ( Hint: Recall trigonometric identities.) 213. Draw graphs of y = x 2 , y = x 6 , and y = x 10 . For y = x n , as n increases, formulate a prediction on the arc length from (0, 0) to (1, 1). Now, compute the lengths of these three functions and determine whether your prediction is correct. 214. Compare the lengths of the parabola x = y 2 and the line x = by from (0, 0) to ⎛ ⎝ b 2 , b ⎞ ⎠ as b increases. What do you notice? 215. x = y 2 from (0, 0) to (1, 1). Show that x = (1/2) y 2 from (0, 0) to (2, 2) is twice as long. Graph both functions and explain why this is so. 216. [T] Which is longer between (1, 1) and (2, 1/2): the hyperbola y =1/ x or the graph of x +2 y =3? Solve for the length of 217. Explain why the surface area is infinite when y =1/ x is rotated around the x -axis for 1≤ x <∞, but the volume is finite.
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