**Unformatted text preview: **Math 21B
Kouba
Discussion Sheet 10 1.) Compute the following improper integrals. 0° 1 5 1 0° 24
. d b. ——— d . M
a) /2 :1:(ln3r)2 a: ) ./0 (/25 — x2 :1: C) /1 21:2 + 5:5 + 2 dm 5 oo 7r/2
8
d.) f 2: dz: e.) / 13:26:53 da: f.) / cscmcota: d1:
0 0 oo oo 1
g.) / ate—5“: dm h.) / :2 d9: i.) / Ina: dx
0 0 33 0 C 3 62.7: 00 e~1/2
j.) / wlna: dx k.) / 2 da: 1.) / 2 d3:
0 0 e a” — 5 0 a: 2.) Use the Comparison Test or Absolute Convergence Test to show that each of the
following improper integrals converges, i.e., is ﬁnite. 0° 1 0° cos :1: ~ sin 23:
a. ___ dac b. ——— dx
l/l m3 + 16 )1, :c + x2
3.) Use Pappus TheOrem (See p. 498, problem 9.) to ﬁnd the centroid (ﬂy) of the triangle with vertices (0,0), (3,0), and (0,4) . HINT : The volume of a cone of radius r
and height h is V : éwrzh . 4.) Find the area of the region bounded by the graphs of 8 = 7r/6 , : 7r/4 , and
7“ : sec6 . 5.) Find the area of the region lying inside the graph of r = 2 + sin0 and outside the
graph of r : c036 . 6.) Find the area of the region lying inside the graph of r : 4sin6 and above the line
7' : cscH . 7.) Compute the arc length of the given curve on the indicated interval. a.) y 2 2:5/4 on the interval [0, 1]
b.) a: : cost + tsint and y : sint ~ tcOst on the interval [7r/6, 7r/4]
b.) r : sin2(6/2) on the interval [0, 7r] 8.) Find the maximum y—value and the maximum x-value on the graph of 7' : 1 — sin9 . ...

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- Winter '09
- Kouba
- Calculus