Sheet9 - Math 21B Kouba Challenge Sheet 9 1 Evaluate the...

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Unformatted text preview: Math 21B Kouba Challenge Sheet 9 1.) Evaluate the following improper integrals. Make careful use of limits and notation. 1 oo 1 1 a.) / —— d3: b. / —— d3: 0 fl ) 1 V; 2 00 1 00 c ) f 2 d3: d.) / W dm fill? +1 0 w5+m3+w+7 °° 1 1 1 e.) f 2 dm f.) / d3: 4 w _‘ 9 _1 Iml 1 g.) / lna: dzr (HINT: You will need integration by parts and L’Hopital’s Rule.) 0 2.) Use the following tests to determine if the following improper integrals converge or diverge— Comparison Test for Convergence, Comparison Test for Divergence, or Absolute Convergence Test. 0° 1 1 1 a. ————— d b, ——————~—-— dx l/l fimm m )1, fimm 1 l c.) / Ina: d3; (HINT: Let f(:c) : nut and consider lim f(:c), lim (5r), 0 1 - $2 1— {132 x—>0+ :c—>1— and the graph of f) 00 - 1 d.) / 811122: dw e.) / C0815 dm (HINT: Consider the maximum and 1 $ 0 CL’ minimum values of y : C083: on the interval [0, 1].) m 00 1 . . f.) / CO” do: g.) f ““2”” dx (HINT: Recall that lim 3‘” = 1 .) 1 93 0 :1: art—>0 :3 0° sina: ' °° , h.) / da: i.) / cos(ez) d3: (HINT: Begin with a u-substitution.) 0 513 0 3.) Consider the region above the x-axis and between the two given circles. Set up integrals for the centroid, (3‘7, 3]), of the region. Use a graphing calculator to evaluate the integrals. Does the centroid lie inside the region ? J ...
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