This article provides step-by-step solutions to Math Quiz #7 problems on area, volume, integrals, power series, and more. Topics covered include finding the area of a region bounded by a graph, evaluating integrals, determining the convergence of improper integrals and series, and using series to evaluate limits.
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QUIZ # 7 Problems (10 questions, 10 points each) 1.Find the area of the region bounded by the graph of𝑓(𝑥)= 2𝑥2+ 8and𝑔(𝑥)= 4𝑥 + 14,at the intervale[−3, 6] Solution – Area of region bounded by the graph is given by, ∫|𝑓(𝑥)− 𝑔(𝑥)|𝑑𝑥 𝑏 𝑎 𝑓(𝑥)= 2𝑥2+ 8 𝑔(𝑥)= 4𝑥 + 14 Therefore, Area, A =∫|2𝑥2+ 8 − 4𝑥 + 14|𝑑𝑥 6 −3 Solving the integral, we get A = 290/3 = 96.67 2.Find the volume of the solid generated by rotating the region bounded by𝑦= 4𝑥 − 𝑥2 about the line𝑥= 5. Solution – Given equation,𝑦= 4𝑥 − 𝑥2 About the line x = 5 means x = 0 to 5 Volume = = 130.9 3.Determine if the improper integral converges and, if so, determine its value:∫1 √8−𝑥 3 8 0𝑑𝑥 Solution – The problem point is the upper limit Solving for, ∫1 √8−𝑥 3 𝑡 0𝑑𝑥=−3(8−𝑡) 2 3 2+ 6 Now Solving for,
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lim 𝑡 →8 −3(8−𝑡) 2 3 2+ 6= 1.61 The limit exists and is finite and so the integral converges and the integral's value is 1.61 4.Evaluate the indefinite integral∫𝑥𝑙𝑛(𝑥)𝑑𝑥 Solution – Applying Integration by parts, u = ln x and v = x Solving,∫𝑥 2𝑑𝑥=𝑥2 4 Therefore,∫𝑥𝑙𝑛(𝑥)𝑑𝑥 5.Evaluate∫4𝑥2+5 √𝑥 3 1𝑑𝑥 Solution: Applying Integration by parts u =4𝑥2+ 5and v =1 √𝑥 Simplifying, Substituting in the above equation, Computing for boundaries
Plugging in x = 1 =58 5 And, Substituting all the values, = 30.66 6.A force of 10 lbs is required to hold a spring stretches 4 inches beyond its natural length. How much work is done to stretches it from its natural length to 5 inches beyond its natural length? Solution – Force = kx 10 = k (1/3)[4 inches = 1/3 feet] K = 30 So, f (x) = 30x Work done W =∫30𝑥 0.416667 0𝑑𝑥= 30[𝑥2 2]= 300.4166672 2 = 2.6 ft.lb 7.Integrate∫𝑒3𝑥 𝑒6𝑥+36𝑑𝑥 Solution:∫𝑒3𝑥 𝑒6𝑥+36𝑑𝑥 Consider u = 3x ∫𝑒𝑢 3(𝑒2𝑢+36)𝑑𝑢 𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑢 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛, 𝑣 =𝑒𝑢
Applying Integral Substitution, v = 6w Taking the constant out, =1 3.1 6tan−1𝑤 =1 3.1 6tan−1𝑒3𝑥 6 Therefore,∫𝑒3𝑥 𝑒6𝑥+36𝑑𝑥=1 18tan−11 6𝑒3𝑥+ C 8.Find the interval of convergence for the power series∑1 2𝑛+2 ∞ 𝑛=1𝑥𝑛+2 Solution: Using the ratio test to find the interval of convergence, We know that, Therefore, Finding, Simplifying, Therefore,
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Also, = 1 = |x| The sum converges for L < 1, therefore solve |x| < 1 For x = 1, there cannot be found a number N which satisfy Cauchy condition, hence it diverges For x = -1= 0 (Converges) Therefore, the interval of convergence for the power series∑1 2𝑛+2 ∞ 𝑛=1𝑥𝑛+2= -1 ≤ x < 1 9.Does the series∑(−1)𝑛𝑥4𝑛 𝑛! ∞ 𝑛=0converges and, if so to what? Solution - Using the ratio test to find the interval of convergence, We know that, Therefore, Computing Solving for, =−1 𝑛
= 0 L < 1 for every x, therefore∑(−1)𝑛𝑥4𝑛 𝑛! ∞ 𝑛=0Converges for all values of x 10.Use a series to evaluate the limitlim 𝑥→0 sin(𝑥)−𝑥+ 1 6𝑥3 𝑥5 Applying L Hospital Rule Applying L Hospital Rule Applying L Hospital Rule Applying L Hospital Rule =cos 0 120=1 120= 0.0083