Is:f(x) dx

f2 f(x) dx + S: f(x) dx - [1 f(x) dx48. If S~f(x) dx = 12 and j; f(x) dx = 3.6, find S~f(x) dx.1m If j~f(x)dx = 37 and S~g(x) dx = 16, findj~[2f(x) + 3g(x)] dx.=eSO. Find J~f(x) dx ifforf(x)x < 3for X"" 35 I. Suppose f has absolute minimum value m and absolute maximumvalue M. Between what two values must S; f(x) dxlie? Which property of integrals allows you to make yourconclusion?52-54 Use the properties of integrals to verify the inequality withoutevaluating the integrals.52. fo l ~ dx~ fal ~ dx[ill 2~ [ ~dx ~ 2J2J2 7T S"/4 J3 7T54. -- ~ cosxdx ~--24 ,,/6 2455. f ..;; dx 56. C 2 __ 1_ dxJo I + x 261- 62 Use properties of integrals, together with Exercises 27 and28, to prove the inequality.'3 2661. j JX4+1 dx ""-I 363. Prove Property 3 of integrals.64. Prove Property 6 of integrals.65. If f is continuous on [a, b], show thatIrf(x) dx I ~ s: I f(x) I dx,,/2 7T 262. fax sin x dx ~ 8IS:"f(x) sin 2x dx I ~ f0 2 " I f(x) I dx67. Letf(x) = 0 if x is any rational number andf(x) = I if x isany irrational number. Show thatfis not integrable on [0, I].68. Letf(O) = 0 andf(x) = l/x if 0 < x ~ 1. Show thatfis notintegrable on [0, I]. [Hint: Show that the first term in theRiemann sum, f(x!) ~x, can be made arbitrarily large.)69.• 1/ ihm L j4[Hint: Consider f(x) = x 4 .)/1--"'00 i=l n'I" I70. Jim - L / 2,,~OO n i~1 1 + (i n)57.f "/3 tan xdx,,/458. S02 (x 3 - 3x + 3) dx71. Find SI 2 x -2 dx. Hint: Choose xt to be the geometric mean ofXi-I and Xi (that is, xt = JXi-IXi ) and use the identity59. [J1h4 dxS O"60. ~ (x - 2 sin x) dxIm(m + I)DISCOVERYPROJECTI. (a) Draw the line y = 2t + I and use geometry to find the area under this line, above thet-axis, and between the vertical lines t = I and t = 3.(b) If x> 1, let A(x) be the area of the region that lies under the line y = 2t + I betweent = I and t = x. Sketch this region and use geometry to find an expression for A(x).(c) Differentiate the area function A(x). What do you notice?2. (a) If x "" -1, letA(x) =t (1 + t 2 ) dt