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Exact Value - Calculus - Exam, Exams of Calculus

National Institute of Industrial EngineeringCalculus

Key points of this past exam of Calculus are: Exact Value, Fundamental Theorem, Definite Integral, Indefinite Integral, Evaluate, Area, Region Bounded, Curve, Determining, Picture

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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MATH106A,B CALCULUS II - PROF. P. WONG
EXAM I - FEBRUARY 1, 2008
NAME:
Instruction: Read each question carefully. Explain ALL your work and
give reasons to support your answers.
Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1. 20
2. 20
3. 20
4. 20
5. 20
Total 100
1
pf3
pf4
pf5

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MATH106A,B CALCULUS II - PROF. P. WONG

EXAM I - FEBRUARY 1, 2008

NAME:

Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.

Problems Maximum Score Your Score

  1. 20
  2. 20
  3. 20
  4. 20
  5. 20 Total 100

1

2 EXAM I - FEBRUARY 1, 2008

1.(10 pts.)(a) Find the exact value (by the Fundamental Theorem of Calculus) of the definite integral ∫ (^) e 1

1 + (ln x)^2 x dx.

(10 pts.)(b) Evaluate the indefinite integral ∫ (^) x √ 1 − x^4

dx.

4 EXAM I - FEBRUARY 1, 2008

  1. (10 pts.)(a) Consider a function h given by the following table. x 1 1.5 2 2.5 3 3.5 4 h(x) -1 2 1 0 -2 3 1 Find R 6 , M 3 using the right-hand sum and the mid-point rule respectively for estimating the definite integral

1 h(x)^ dx.

(10 pts.)(b) Recall that the error committed by using the left hand sum approximation Ln is less than or equal to K^1 ·( 2 bn− a)^2 where |f ′(x)| ≤ K 1 for some constant K 1 over the interval [a, b]. Use this result to give an upper bound for the error committed by L 10 for

I =

0

(sin x)ex^ dx.

MATH106A,B CALCULUS II - PROF. P. WONG 5

  1. Let R be the region bounded by the curve y = − 1 − x^2 , the line x = 1, the x-axis and the y-axis. (15 pts.) Find the exact volume of the solid obtained from rotating the region R around the x-axis.

A

B y=−1−x

x=

R

2

(5 pts.) SET UP (do not evaluate) a definite integral representing the arc length of the portion of the curve from A to B.