The energy used was approximately 15,840 megawatt-hours. A note on units How did we know what units to use for energy in Example 7? The integral f* P(t) dt is defined as the limit of sums of terms of the form P(t*) At. Now P(t;) is measured in megawatts and At is measured in hours, so their product is measured in megawatt-hours. The same is true of the limit. In general, the unit of measurement for faf(x) dx is the product of the unit for f(x) and the unit for x. 4.4 EXERCISES 1-4 Verify by differentiation that the formula is correct. 11 . ( 1 + vx + * dx dx = _ V1+x2 + c Vx 1. 72 1+ x 2 X 2. [cos x dx = Ex + 4 sin 2x + C 12 . J ( 1 2 + 1 + ) du 14. sec t (sec t + tan t) dt 3. tan'x dix = tan x - x + C 13 . ( 2 + tan ' 0 ) do 4. fxva+ bx dx = 2 15 . (1 1 - sin't at 16. sin 2x 156 2 (3bx - 2a ) (a + bx ) 3 /2 + C sin't sin x 5-16 Find the general indefinite integral. 17-18 Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. 5 . ( 2 13 + 7 x 25 ) dx 17 (cos x + {x) dx 18 . (1 - x 2 ) 2 dx 6. Vx dx 19-42 Evaluate the integral. 7. (5 + 3x2 + 2x) dx 19 . ( x 2 - 3 ) dx 20 . ( 4x3 - 3x 2 + 2x ) dx 8. (16 - 215 - 13 + 7) du 27 . [ o ( 2 # 4 + 413 - 1 ) de 9 . (u + 4 ) ( 2u + 1 ) du 10 . ( Vi (1 2 + 3+ + 2 ) dt 22 . (1+ 6w 2 - 10 w 4 ) du33. (7/4 1 + cos20 do cos20 49. I 34. (#/3 sin 0 + sin 0 tan20 Jo do sec-0 50. 35. (8 2 + t dt 3/ 7 2 36. Vu(u - Vu ) du 37. ( V x5 + 8/ x4 ) dx 38 . ( 1 + x 2 ) 3 dx 51. I 39 [ lx - 3/ dx 40. 2 |2x - 1/ dx 52. I 47 . [2 ( x - 21 x1) dx 42 | sin x | dx 53. I 43. Use a graph to estimate the x-intercepts of the curve 54. I y = 1 - 2x - 5x4. Then use this information to estimate the area of the region that lies under the curve and above the x-axis. 44. Repeat Exercise 43 for the curve y = 2x + 3x4 - 2x6. 55-56 45. The area of the region that lies to the right of the y-axis and a part to the left of the parabola x = 2y - y2 (the shaded region (b) th inter in the figure) is given by the integral Jo (2y - y?) dy. (Turn your head clockwise and think of the region as lying below 55. v the curve x = 2y - y' from y = 0 to y = 2.) Find the area 56. of the region. 57-58 2 ity are x = 2y - y2 veloc time i 57. a 58. 46. The boundaries of the shaded region in the figure are the 59 y-axis, the line y = 1, and the curve y = Vx . Find the area of this region by writing x as a function of y and integratingof the curve 54. If the units for x are feet and the units for a(x) are pounds nation to estimate per foot, what are the units for da/ dx? What units does curve and above a(x) dx have? + 3.x4 - 2.16. 55-56 The velocity function (in meters per second) is given for t of the y-axis and a particle moving along a line. Find (a) the displacement and he shaded region (b) the distance traveled by the particle during the given time interval. y - y') dy. (Turn on as lying below 55. v( 1 ) = 31 - 5 , 05153 2.) Find the area 56, v(1) = 12 - 21 - 3, 2 5 154 57-58 The acceleration function (in m/s2) and the initial veloc- ity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval. 57. a(t) = 1+ 4, 0(0) = 5, 0 t= 10 58. a(t) = 2t + 3, v(0) = -4, 0=1=3 figure are the 59. The linear density of a rod of length 4 m is given by . Find the area p(x) = 9 + 2 vx measured in kilograms per meter, where and integrating x is measured in meters from one end of the rod. Find the total mass of the rod