564
Chapter 5 | Integration
π /4
198. Suppose that the number of hours of daylight on a given day in Seattle is modeled by the function −3.75cos ⎛ ⎝ πt 6 ⎞ ⎠ +12.25, with t given in months and t =0 corresponding to the winter solstice. a. What is the average number of daylight hours in a year? b. At which times t 1 and t 2 , where 0≤ t 1 < t 2 <12, do the number of daylight hours equal the average number? c. Write an integral that expresses the total number of daylight hours in Seattle between t 1 and t 2 . d. Compute the mean hours of daylight in Seattle between t 1 and t 2 , where 0≤ t 1 < t 2 <12, and then between t 2 and t 1 , and show that the average of the two is equal to the average day length. 199. Suppose the rate of gasoline consumption over the course of a year in the United States can be modeled by a sinusoidal function of the form ⎛ ⎝ 11.21−cos ⎛ ⎝ πt 6 ⎞ ⎠ ⎞ ⎠ ×10 9 gal/mo. a. What is the average monthly consumption, and for which values of t is the rate at time t equal to the average rate? b. What is the number of gallons of gasoline consumed in the United States in a year? c. Write an integral that expresses the average monthly U.S. gas consumption during the part of the year between the beginning of April ( t =3) and the end of September ⎛ ⎝ t =9). 200. Explain why, if f is continuous over ⎡ ⎣ a , b ⎤ ⎦ , there is at least one point c ∈ ⎡ ⎣ a , b ⎤ ⎦ such that
184. ∫
sec 2 θdθ
0
π /4
185. ∫
sec θ tand θ
0
π /4
186. ∫
csc θ cot θdθ
π /3
π /2
187. ∫
csc 2 θdθ
π /4
2 ⎛
⎞ ⎠ dt
188. ⌠
⎝ 1
− 1 t 3
⌡ 1
t 2
−1 ⎛
⎞ ⎠ dt
189. ⌠
⎝ 1
− 1 t 3
⌡ −2
t 2
In the following exercises, use the evaluation theorem to express the integral as a function F ( x ). 190. ∫ a x t 2 dt 191. ∫ 1 x e t dt 192. ∫ 0 x cos tdt 193. ∫ − x x sin tdt In the following exercises, identify the roots of the integrand to remove absolute values, then evaluate using the Fundamental Theorem of Calculus, Part 2. 194. ∫ −2 3 | x | dx 195. ∫ −2 4 | t 2 −2 t −3 | dt 196. ∫ 0 π |cos t | dt 197. ∫ − π /2 π /2 | sin t | dt
b − a ∫ a b
f ( c ) = 1
f ( t ) dt .
201. Explain why, if f is continuous over ⎡ ⎣ a , b ⎤ ⎦ and is not equal to a constant, there is at least one point M ∈ ⎡ ⎣ a , b ⎤ ⎦
b − a ∫ a b
such that f ( M ) = 1
f ( t ) dt and at least one point
b − a ∫ a b
m ∈ ⎡
⎤ ⎦ such that f ( m ) < 1
f ( t ) dt .
⎣ a , b
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