MATH 221 Final: MATH 221 KSU Final Exam f06
15 Feb 2019
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CALCULUS II - FINAL EXAM
December 13, 2006
Show all work for full credit. No books, notes or calculators are permitted. The point value of each
problem is given in the left-hand margin. You have 1 hour and 50 minutes.
Zsec x dx = ln |sec x+ tan x|+C
Zcsc x dx =−ln |csc x+ cot x|+C
Zdx
√a2−x2= arcsin(x
a) + C
Zdx
a2+x2=1
aarctan(x
a) + C
Zdx
x√x2−a2=1
aarcsec(|x|
a) + C
Z√a2−u2du =1
2u√a2−u2+a2arcsin u
a+C,
Z√u2±a2du =1
2u√u2±a2±a2ln |u+√u2±a2|+C
Centroid for the region trapped between y=f(x) , y=g(x) , a≤x≤b, (with ρ= 1 )
Mx=1
2Rb
af(x)2−g(x)2dx ,My=Rb
ax(f(x)−g(x)) dx
Maclaurin Series: ex=P∞
n=0 xn
n!
ln(1 + x) = P∞
n=1
(−1)n+1xn
n
sin x=P∞
n=0
(−1)nx2n+1
(2n+1)!
cos x=P∞
n=0
(−1)nx2n
(2n)!
page 2 of 8
1. Evaluate the following integrals.
(12) a) Zln(x)
x2dx
(12) b) Zx2dx
√1−x2
(12) c) Zx3+ 2
x2−xdx
page 3 of 8
2. Let R be the region trapped between y= cos xand y= sin xwith 0 ≤x≤π/4 .
(6) a) Find the area of the region R.
(8) b) Find y, the ycoordinate of the centroid of R. (Do not calculate x. )
Hint: cos(2x) = cos2(x)−sin2(x) .
3. Evaluate the following limits or indicate that they diverge. Show all work.
(8) a) lim
x→0
e3x−1−3x
xsin(x)
(8) b) lim
x→∞ x1/x
Document Summary
The point value of each problem is given in the left-hand margin. Z sec x dx = ln| sec x + tan x| + c. Z csc x dx = ln| csc x + cot x| + c. ) + c dx a2 + x2 = Z a2 u2 du = arcsec(|x| a. 2 (cid:18)u a2 u2 + a2 arcsin u a(cid:19) + c , 2 (cid:16)u u2 a2 a2 ln|u + u2 a2|(cid:17) + c. Centroid for the region trapped between y = f (x) , y = g(x) , a x b , (with = 1 ) 2 r b a f (x)2 g(x)2 dx , my = r b a x(f (x) g(x)) dx n=0 xn n! n=1 n ( 1)n+1xn. Hint: cos(2x) = cos2(x) sin2(x) : evaluate the following limits or indicate that they diverge. Show all work. (8) a) lim x 0 e3x 1 3x x sin(x) (8) b) lim x x1/x (10) 4.
