Problem 60E: In the following exercises, express the limits as integrals. 60. limni=1n(xi*)x over [1, 3] Problem 61E: In the following exercises, express the limits as integrals. 61. limni=1n(5( x i * )23( x i * )3)x... Problem 62E: In the following exercises, express the limits as integrals. 62. limni=1nsin2x(2xi*)x over [0, 1] Problem 63E: In the following exercises, express the limits as integrals. 63. limni=1ncos2x(2xi*)x over [0, 1] Problem 64E: In the following exercises, given Ln or Rn as indicated, express their limits as n as definite... Problem 65E: In the following exercises, given Ln or Rn as indicated, express their limits as n as definite... Problem 66E: In the following exercises, given Ln or Rn as indicated, express their limits as n as definite... Problem 67E: In the following exercises, given Ln or Rn as indicated, express their limits as n as definite... Problem 68E: In the following exercises, given Ln or Rn as indicated, express their limits as n as definite... Problem 69E: In the following exercises, given Ln or Rn as indicated, express their limits as n as definite... Problem 70E: In the following exercises, evaluate the integrals of the functions graphed using the formulas for... Problem 71E: In the following exercises, evaluate the integrals of the functions graphed using the formulas for... Problem 72E: In the following exercises, evaluate the integrals of the functions graphed using the formulas for... Problem 73E: In the following exercises, evaluate the integrals of the functions graphed using the formulas for... Problem 74E: In the following exercises, evaluate the integrals of the functions graphed using the formulas for... Problem 75E: In the following exercises, evaluate the integrals of the functions graphed using the formulas for... Problem 76E: In the following exercises, evaluate the integral using area formulas. 76. 03(3x)dx Problem 77E: In the following exercises, evaluate the integral using area formulas. 77. 23(3x)dx Problem 78E: In the following exercises, evaluate the integral using area formulas. 78. 33(3|x|)dx Problem 79E: In the following exercises, evaluate the integral using area formulas. 79. 06(3|x3|)dx Problem 80E: In the following exercises, evaluate the integral using area formulas. 80. 224x2dx Problem 81E: In the following exercises, evaluate the integral using area formulas. 81. 154 ( x3 )2dx Problem 82E: In the following exercises, evaluate the integral using area formulas. 82. 01236 ( x6 )2dx Problem 83E: In the following exercises, evaluate the integral using area formulas. 83. 23(3|x|)dx Problem 84E: In the following exercises, use averages of values at the left (L) and tight (R) endpoints to... Problem 85E: In the following exercises, use averages of values at the left (L) and tight (R) endpoints to... Problem 86E: In the following exercises, use averages of values at the left (L) and tight (R) endpoints to... Problem 87E: In the following exercises, use averages of values at the left (L) and tight (R) endpoints to... Problem 88E: Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,... Problem 89E: Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,... Problem 90E: Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,... Problem 91E: Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,... Problem 92E: Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,... Problem 93E: Suppose that 04f(x)dx=5 and 02f(x)dx=3 , and 04g(x)dx=1 and 02g(x)dx=2 . In the following exercises,... Problem 94E: In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals.... Problem 95E: In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals.... Problem 96E: In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals.... Problem 97E: In the following exercises, use the identity AAf(x)dx=A0f(x)dx+0Af(x)dx to compute the integrals.... Problem 98E: In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals. 98.... Problem 99E: In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals. 99.... Problem 100E: In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals.... Problem 101E: In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals.... Problem 102E: In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals.... Problem 103E: In the following exercises, given that 01xdx=12,01x2dx=13 , and 01x3dx=14 compute the integrals.... Problem 104E: In the following exercises, use the comparison theorem. 104. Show that 03(x26x+9)dx0 . Problem 105E: In the following exercises, use the comparison theorem. 105. Show that 23(x3)(x+2)dx0 . Problem 106E: In the following exercises, use the comparison theorem. 106. Show that 011+x3dx011+x2dx . Problem 107E: In the following exercises, use the comparison theorem. 107. Show that 121+xdx121+x2dx . Problem 108E: In the following exercises, use the comparison theorem. 108. Show that 0/2sintdt4 . (Hint: sint2t... Problem 109E: In the following exercises, use the comparison theorem. 109. Show that /4/4costdt2/4 . Problem 110E: In the following exercises, find 1112 average value fave of f between a and b, and find a point c,... Problem 111E: In the following exercises, find 1112 average value fave of f between a and b, and find a point c,... Problem 112E: In the following exercises, find 1112 average value fave of f between a and b, and find a point c,... Problem 113E: In the following exercises, find 1112 average value fave of f between a and b, and find a point c,... Problem 114E: In the following exercises, find 1112 average value fave of f between a and b, and find a point c,... Problem 115E: In the following exercises, find 1112 average value fave of f between a and b, and find a point c,... Problem 116E: In the following exercises, approximate the average value using Riemann sums L100 and R100. How does... Problem 117E: In the following exercises, approximate the average value using Riemann sums L100 and R100. How does... Problem 118E: In the following exercises, approximate the average value using Riemann sums L100 and R100. How does... Problem 119E: In the following exercises, approximate the average value using Riemann sums L100 and R100. How does... Problem 120E: In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,... Problem 121E: In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,... Problem 122E: In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,... Problem 123E: In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,... Problem 124E: In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,... Problem 125E: In the following exercises, compute the average value using the left Riemann sums LN for N = 1, 10,... Problem 126E: Show that the average value of sin2t over [0, 2 ] is equal to 1/2 Without further calculation,... Problem 127E: Show that the average value of cos2t over [0, 2 ] is equal to 1/2. Without further calculation,... Problem 128E: Explain why the graphs of a quadratic function (parabola) p(x) and a linear function l (x) can... Problem 129E: Suppose that parabola p(x)=ax2+bx+c opens downward (a < 0) and has a vertex of y=b2a0 . For which... Problem 130E: Suppose [a, b} can be subdivided into subintervals a=a0a1a2...aN=b such that either f0 over [ai1,ai]... Problem 131E: Suppose f and g are continuous functions such that cdf(t)dtcdg(t)dt for every subinterval [c, d] of... Problem 132E: Suppose the average value of f over [a, b] is 1 and the average value of f over [b, c] is 1 where a... Problem 133E: Suppose that [11. b] can be partitioned, taking a=a0a1...aN=b such that the average value of f over... Problem 134E: Suppose that for each i such that 1iN one has i1if(t)dt=i . Show that 0Nf(t)dt=N( N+1)2 . Problem 135E: Suppose that for each i such that 1iN one has i1if(t)dt=i2 . Show that 0Nf(t)dt=N( N+1)( 2N+1)6 . Problem 136E: [T] Compute the left and right Riemann sums L10 and R10, and their average L10+R102 for f(t)=t2 over... Problem 137E: [T] Compute the left and right Riemann sums, L10 and R10, and their average L10+R102 for f(t)=(4t2)... Problem 138E: If 151+t4dt=41.7133... , what is 151+u4du ? Problem 139E: Estimate 01tdt using the left and light endpoint sums, each with a single rectangle. How does the... Problem 140E: Estimate 01tdt by comparison with the area of a single rectangle with height equal to the value of t... Problem 141E: From the graph of sin(2(x) shown: a. Explain why 01sin(2t)dt=0 . b. Explain why, in general,... Problem 142E: If f is 1-periodic (f(t+1)=f(t)) , odd, and integrable over [0, l], is it always true that... Problem 143E: If f is 1-periodic and 01f(t)dt=A , is it necessarily true that a1+af(t)dt=A for all A? format_list_bulleted