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Math 106A Exam 1 - February 3, 2006, Exams of Calculus

National Institute of Industrial EngineeringCalculus

The instructions and problems for exam 1 of math 106a, including integrals, area calculations, and differential equations. Students are required to show their work and provide solutions.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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NAME:
Math 106A - Exam 1 - February 3, 2006
INSTRUCTIONS: Show all of your work and circle your solutions. Cross out any unnecessary work.
1. Integrate the following. If you use a formula from the table of integrals, indicate which formula you’re using.
(5 pts. each)
(a) Zx2cos (x3)dx
(b) Z1
0
arctan (x)dx.
(c) Z3x2+x+ 3
x3+xdx
pf3
pf4
pf5

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NAME:

Math 106A - Exam 1 - February 3, 2006

INSTRUCTIONS: Show all of your work and circle your solutions. Cross out any unnecessary work.

  1. Integrate the following. If you use a formula from the table of integrals, indicate which formula you’re using. (5 pts. each)

(a)

x^2 cos (x^3 )dx

(b)

0

arctan (x)dx.

(c)

3 x^2 + x + 3 x^3 + x

dx

(d)

1

2 x + 3 (x + 2)^2

dx.

(e)

e

√x cos(

x) √ x

dx

  1. (8 pts.) Calculate the area of the region bounded by the function y =

x, the x-axis, and the line y = x − 2. (Hint: Integrate in terms of y.)

  1. (5 pts. each) Let I =

1 ln^ xdx. (a) Calculate M 10. (Use your calculator.) Explain why this will be an overestimate.

(b) Using the error bound theorem, how accurate (to six decimals) is M 10 as an estimate of I?

(c) Solve the integral I and give the actual error (to six decimals) of M 10.

(d) Using the error bound theorem, how many subintervals would we need to use with Simpson’s rule to guarantee accuracy to within ± 0 .00001?

  1. (8 pts.) Let f (x) = 23 x^3 /^2 + 5. Find the length of the curve y = f (x) from x = 0 to x = 3.
  2. (8 pts.) Find the volume of the solid of revolution formed when the region bounded by the curve y = x^2 and the x-axis between x = 0 and x = 2 is rotated around the y-axis.