The idea of the integralChapter 8: Practice/review problemsThe collection of problems listed below comprises questions taken from previous MA123 exams.[1]. Estimate the area under the graph of f (x) = x2+ 2 on the interval [0, 2] by dividing the interval into fourequal parts. Use the right endpoint of each interval as a sample point.(a) 11.75 square units (b) 5.71875 square units (c) 9.5 square units(d) 5.75 square units (e) 7.75 square units[2].Estimate the area under the graph of y = x2+ 2x + 3for x between −2 and 2. Use a partition that consists of4 equal subintervals of [−2, 2] and use th e right endpoint ofeach subinterval as a sample point.(a) 22(b) 23(c) 24(d) 25(e) 26xy−2 −10 1 2[3]. A train starts from rest (velocity equal to 0 miles per hour) at 12:00 noon. The velocity increases at aconstant rate until 12:15 when the velocity equals 64 miles per hour. How far does the train travel from12:00 to 12:15?(a) 7 (b) 8 (c) 9 (d) 10 (e) 11[4]. Use a calculator to estimate the integralZ.25.12xdxUse three (3) subintervals and the left endpoint of each subinterval to determine the height of the rectanglesused in the approximation. The approximate value of the integral is(a) .166 (b) .168 (c) .172 (d) .174 (e) .178[5]. Use a calculator to estimate the integralZ2.252log(x) dxUse five (5) subintervals and the left endpoint of each subinterval to determine the height of the rectanglesused in the approximation. The approximate value of the integral is(a) .131 (b) .128 (c) .113 (d) .104 (e) .08117[6]. Use a calculator to estimate the integralZ21ln(x) dx.Use four subintervals and the right endpoint of each subinterval to determine the height of the rectanglesused in the approximation. The approximate value of the integral is(a) 0.218 (b) 0.297 (c) 0.352 (d) 0.470 (e) 0.521[7].Estimate the area under the graph of y = 2x2for x between1 and 5. Use a partition that consists of 4 equal subintervals of[1, 5] and use the right endpoint of each subinterval as a samplepoint.(a) 92(b) 94(c) 96(d) 102(e) 108xy0 1 2 3 4 5[8]. Suppose you estimate the integralZ2010(1 + x)2dxby the sum of the areas of 10 r ectangles of equal base length. Use the right endpoint of each base todetermine the height. What is the area of the first (left most) rectangle?(a) 144 (b) 244 (c) 341 (d) 441 (e) 541[9]. Suppose you estimate the area under the graph of f(x) = x3from x = 5 to x = 25 by adding the areas ofrectangles as follows: partition the interval into 20 equal subintervals and use the right endpoint of eachinterval to determine the height of the rectangle. What is the area of the 11th rectangle?(a) 1000 (b) 1331 (c) 2744 (d) 3375 (e) 4096[10]. You want to estimate the integralZ30101xdx as the sum of areas of rectangles. You break the interval[10, 30] into 20 subintervals of equal length. If you use the left endpoint of each subinterval to determinethe height of each r ectangle, which estimate is correct?(Hint: Draw a picture!)(a)Z30101xdx ≥110+111+112+ ··· +129(b)Z30101xdx ≤110+111+112+ ··· +129(c)Z30101xdx ≤111+112+ ··· +129+130(d)Z30101xdx ≥111+112+ ··· +129+130(e)Z30101xdx ≤110+112+114+ ··· +128+130118[11]. Suppose you estimate the integralZ2010x2dx by the sum of th e areas of 50 rectangles of equal baselength. Use the left endpoint of each base to determine the height. What is the area of the first (leftmost)rectangle?(a) 20 (b) 30 (c) 40 (d) 50 (e) 60[12]. Evaluate the sum10Xk=4(1 + k)(a) 56 (b) 60 (c) 63 (d) 73 (e) 74[13]. Write the sum 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 in summation notation asNXk=2(A + 2k)What are the values of A and N?(a) A = 1, N = 10 (b) A = 2, N = 10 (c) A = 1, N = 9(d) A = 2, N = 9 (e) A = 3, N = 9[14]. Evaluate the sum4Xk=2(5k + 1)(a) 112 (b) 66 (c) 29 (d) 64 (e) 48[15]. Evaluate the sum6Xk=3(k2− 1)(a) 35 (b) 60 (c) 82 (d) 98 (e) 122[16]. Evaluate the sum6Xk=2(k2− k).(a) 60 (b) 63 (c) 67 (d) 70 (e) 72[17]. Suppose you estimate the integralZ62f(x) dxby evaluating the sumnXk=1(∆x)f(2 + k∆x).If you use ∆x = .2, what value should you use for n?(a) 25 (b) 10 (c) 30 (d) 20 (e) 15119[18]. You make two estimates using rectangles for the integralZ10(1 − x2)dxThe first estimate uses 50 equal length subintervals and the left endpoint of each subinterval. Th e secondestimate uses 50 equal length subintervals and the right endpoint of each subinterval. What is thedifference between the two estimates (first minus second)?(a)850(b)650(c)450(d)250(e)150[19]. The integralZ61x3dxis computed aslimn→∞nXk=1An1 + k ·An3What is the value of A?(a) 8 (b) 5 (c) 7 (d) 6 (e) 4[20]. Suppose you estimate the integralZ62f(x) dxby adding the areas of n rectangles of equal base length, and you use the right end point of each s ubintervalto determine the height of each rectangle. If the sum you evaluate is w ritten asnXk=1An· fB +Ank,what are A and B?(a) A = 2, B = 4 (b) A = 4, B = 4 (c) A = 4, B = 2(d) A = 2, B = 2 (e) None of th e above[21]. Suppose that you estimate the integralZ82f(x) dxby evaluating a sumnXk=1∆x · f(2 + k · ∆x).If you use 12 intervals of equal length, what value should you use for ∆x?(a) 0.1 (b) 0.2 (c) 0.3 (d) 0.4 (e) 0.5120[22]. Suppose th at the integralZ111f(x) dx is estimated by the s umNXk=1f(a + k∆x) · ∆x. The termsin the sum equal areas of rectangles obtained by using right endpoints of the subintervals of length ∆xas sample points. If N = 20, then what is ∆x?(a) .05 (b) .1 (c) .5 (d) 1 (e) Cannot be determined[23]. Suppose th at the integralZ522f(x) dx is estimated by the s umNXk=1f(a + k∆x) · ∆x. The termsin the sum equal areas of rectangles obtained by using right endpoints of the subintervals of length ∆xas sample points. If f (x) =1x2and N = 50, then find the area of the second rectangle.(a) 1/16 (b) 1/9 (c) 1/8 (d) 1/4 (e) 1/2[24]. Suppose that the integralZ126√x dx is estimated by the sumNXk=1p(a + k∆x) · ∆x, where∆x = .2 and N = 30. The terms in the sum equal areas of rectangles obtained by u sing right endpointsof the subintervals of length ∆x as sample points. What is a?(a) 2 (b) 3 (c) 4 (d) 5 (e)
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