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using a regular partition of[1,4] into three equal subintervals.10
(b) CalculateU4, the upper sum forfusing a regular partition of[1,4] into four equal subintervals.60. Letf(x) = 4-x2.For any positive integern, calculate lower andupper estimates for the areaAunder the graph ofy=f(x) over theinterval [0,2] using a regular partition of this interval intonequalsubintervals (as was done in lectures fory=x2over [0, b]). Hence findA.61. Letf(x) = 10-2x,x∈[1,4].(b) Using formulas from lectures, find an expression forNXi=1f(xi) Δxthat does not involve the summation symbol Σ.[Suggestion:simplifyf(xi) first.](c) Find the limiting value of your expression in (b) asNbecomeslarger and larger.62. CalculateU4andL4for the integralsin(x)dx, giving your answeraccurate to 3 decimal places.(You will need to use a scientific orgraphing calculator.)Z363. Use upper and lower sums to evaluate(5-2x)dx.1Properties of the Definite Integral64. Given thatZ30(2f(x)-3g(x))dx= 3 andZ30(f(x) +g(x))dx= 4,findZ30(f(x)-g(x))dx.65. Supposef:R→Ris a continuous function satisfying-1f(x)dx= 1andZ10f(x)dx= 3. FindZ31f(x)dxin each of the following cases:(a)f(x) = 2 for allx∈[-1,0];(b)fis anodd function(i.e., for anyx,f(-x) =-f(x));(c)fis aneven function(i.e., for anyx,f(x) =f(-x)).Z3Definite Integrals and Areas66. LetG(x) be given byG(x) =Zx-xp1-t2dt11
forx∈[0,1]. FindG12.Hint: Try sketchingy=p1-x2and using the area interpretation ofthe integral.67. It is known that the area under the graph ofy= sin(x) over theinterval [0, π] is exactly 2.By considering graphs, explain why thearea under the graph ofy= arcsin(x) over [0,1] isπ/2-1.Hint: recall that ify=f(x)thenx=f-1(y).Antiderivatives and The Fundamental Theorem Of Cal-culus68. Calculate the following, showing your working:(a)F0(2)ifF(x) =Zx-1p1 +t3dt.(b)G(2)ifG(x) =Zx1ddthp1 +t3idt.(c)Z10f(t)dtwheref(t) =ddtcos(πt3) .69. Below is the graph of a functionf. For each of the parts below, theanswer is aninteger. Given this fact, find the following:(a)Z20f(x)dx(b)Z42f(x)dx(c)Z21f0(x)dx(d)F0(2), whereF(x) =Zx1f(t)dt(e)Z64|f(x)|dx(f)U3forZ60f(x)dx.Indefinite integrals and evaluation of definite integralsusing antiderivatives70. Find an antiderivative for each of the following functions:(a)f(x) = 4x3-2x+ 7(b)g(x) =-4x3+ 2 sin(x)12