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KOCยธ UNIVERSITY
MATH 106 - CALCULUS
Midterm III May 9, 2007
Duration of Exam: 75 minutes
INSTRUCTIONS: Calculators may not be used on the test. No books, no notes, and no talking allowed. You must always explain your answers and show your work to receive full credit. Use the back of these pages if necessary. Print and sign your name, and indicate your section below.
Surname, Name: โโโโโโโโโโโโโโโโโ
Signature: โโโโโโโโโโโโโโโโโโโโ
Section (Check One):
Section 1 - 11:30 โโ Section 2 - 14:30 โโ
PROBLEM POINTS SCORE
TOTAL 106
- (a) (10 points) The region between the curve y = cos x, 0 โค x โค ฯ 4 and the x-axis is revolved about the x-axis to generate a solid. Find its volume.
(b) (10 points) Find the length of the curve y = 2x^3 /^2 + 1 from x = 0 to x = 1.
- Calculate the following integrals.
(a) (10 points)
x sin x dx
(b) (10 points) dx (x + 1)(x^2 + 1)
(c) (10 points)
โโ
2 x dx (x^2 + 1)^2
- (a) Determine whether the following sequences {an} converges or diverges. If the sequence converges, find its limit.
(i) (8 points) an = n
10 n
(ii) (8 points) an =
n
)n
(b) Determine whether the following series converges or diverges. If the series converges, find its sum.
(i) (6 points)
โ^ โ
n=
2 n^3 + n โ 1 3 n^3 โ n^2 + 1
(ii) (8 points)
โ^ โ
n=
3 n+ 4 n
