166 MA NAME STUDENT ID
EXAM 2
Spring 2004
Page 1/4
Page 1 Page 2 Page 3 Page 4
/12 /36 /30 /22 /100
RECITATION INSTRUCTOR RECITATION TIME DIRECTIONS
TOTAL
1. Write your name, student ID number, recitation instructor's name and recitation time in the space provided above. Also write your name at the top of pages 2, 3, and 4 . 2. The test has four (4) pages, including this one. 3. Write your answers in the boxes provided. 4. You must show sufficient work to justify all answers. Correct answers with inconsistent work may not be given credit. 5. Credit for each problem is given in parentheses in the left hand margin. 6. No books, notes or calculators may be used on this test. Find the integrals in problems 1-5. (6) 1.(a) sin3 x cos2 xdx.
(6) 2.
tan3 x sec3 xdx.
MA 166
1
Exam 2 (tan-1 x)2 dx. 1 + x2
Spring 2004
Name
Page 2/4
(6) 3.
0
(15) 4.
x2 (a2 - x2 ) 2
3
dx, where a is a positive constant.
(15) 5.
3x2 - 1 dx (x - 1)(x2 + 1)
MA 166
Exam 2
Spring 2004
Name
Page 3/4 1 dx as an integral of x x+1
(6) 6. a Make substitution u = . . . to express the integral a rational function in u. Do not evaluate the integral.
1 dx = x x+1
du
(12) 7. Determine whether each integral is convergent or divergent. Find its value if it is convergent. Important: You must show clearly how limits are involved.
(a)
1
ln x dx x
1
(b)
0
1 dx 1 - x2
(12) 8. Find the length of the curve y = ln(sec x), 0 x . 4 d ln(sec x + tan x) = sec x]. [Note: dx
MA 166
Exam 2
Spring 2004
Name
Page 4/4
(12) 9. Consider the lamina bounded by the curves y = x3 , y = 1, x = 0 and with density = 1. Find the following: (a) The mass m of the lamina. m= (b) The moment My of the lamina about the yaxis. My = (c) The moment Mx of the lamina about the xaxis. Mx = (d) The center of mass (x, y) of the lamina. (x, y) = (10) 10. Determine whether the sequence converges or diverges. If it converges, find the limit. (You need not show work for this problem). 3n (a) an = n+2 4
(b)
3 + (-1)n n2
(c) an =
n ln n
n3 (d) an = 1 + n2
(e) an =
(-1)n n! (n + 1)!