594
Chapter 5 | Integration
⌠ ⌡ ⎮⎮ 0 1
318. [T] The following graph is of a function of the form f ( t ) = a sin( nt )+ b sin( mt ). Estimate the coefficients a and b , and the frequency parameters n and m . Use these estimates to approximate ∫ 0 π f ( t ) dt .
1−2 t ⎛ ⎝ 1+ ⎛ ⎝ t − 1 2
2 ⎞ ⎠
dt
309.
⎞ ⎠
⎛ ⎝ ⎜ ⎛
3 ⎞ ⎠
310. ⌠ ⌡ 0 π
⎞ ⎠
⎛ ⎝ t − π 2
⎞ ⎠ dt
⎟ cos
⎝ t − π 2
sin
2 (1− t )cos( πt ) dt
311. ∫
0
3 π /4
312. ∫
sin 2 t cos tdt
π /4
319. [T] The following graph is of a function of the form f ( x ) = a cos( nt )+ b cos( mt ). Estimate the coefficients a and b and the frequency parameters n and m . Use these estimates to approximate ∫ 0 π f ( t ) dt .
313. Show that the average value of f ( x ) over an interval ⎡ ⎣ a , b ⎤ ⎦ is the same as the average value of f ( cx ) over the interval ⎡ ⎣ a c , b c ⎤ ⎦ for c >0.
t
314. Find the area under the graph of f ( t ) =
a
⎛ ⎝ 1+ t 2
⎞ ⎠
between t =0 and t = x where a >0 and a ≠1 is fixed, and evaluate the limit as x →∞. 315. Find the area under the graph of g ( t ) = t ⎛ ⎝ 1− t 2 ⎞ ⎠ a between t =0 and t = x , where 0< x <1 and a >0 is fixed. Evaluate the limit as x →1. 316. The area of a semicircle of radius 1 can be expressed as ∫ −1 1 1− x 2 dx . Use the substitution x =cos t to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral. 317. The area of the top half of an ellipse with a major axis that is the x -axis from x = a to a and with a minor axis that is the y -axis from y =− b to b can be written as ⌠ ⌡ − a a b 1− x 2 a 2 dx . Use the substitution x = a cos t to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.
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