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Exam 1 for Math 106c: Integration Problems and Approximations, Exams of Calculus

The february 6, 2004 exam for math 106c, focusing on integration problems and approximations. Students are required to compute definite and improper integrals using given formulas, measure rates of water drainage, and evaluate the convergence of improper integrals. They must also explain the meaning and reasoning behind certain integrals and approximations.

Typology: Exams

2012/2013

Uploaded on 03/20/2013

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Math 106c Name :
Exam 1 February 6, 2004
Show all your work. On each problem you must write down enough details of what you have done that
your method is understandable.
1. (42 pts. – 7 pts. each) Compute the following integrals. For definite integrals, give an exact answer
rather than a numerical approximation. You may use the table of integrals as long as you cite the
number of the formula(s) you use.
(a) Z√π
0
xsin(x2)dx
(b) Z1
x(x−2)(x+3)dx
(c) Z5
1
z√5−zdz
(d) Zarctan sds
1
pf3
pf4

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Math 106c Name : Exam 1 February 6, 2004

Show all your work. On each problem you must write down enough details of what you have done that your method is understandable.

  1. (42 pts. – 7 pts. each) Compute the following integrals. For definite integrals, give an exact answer rather than a numerical approximation. You may use the table of integrals as long as you cite the number of the formula(s) you use.

(a)

∫ √π

0

x sin(x^2 ) dx

(b)

x(x − 2)(x + 3)

dx

(c)

1

z

5 − z dz

(d)

arctan s ds

(e)

x^2 + 4x + 5 dx

(f)

5

t ln t

dt

  1. (15 pts. – 5 pts. each) Measurements are made of the rate at which water is draining from a container at various times, and recorded in the following table:

t (min) 0 .5 1 1.5 2 r(t) (liters/min) 17 16 14 10 3

(a) In a sentence or two, explain why someone might want to calculate

0

r(t)dt. What is the meaning of this integral?

(b) Give the best approximation you can to the value of

0

r(t)dt. You should show all your work, so that your method is clear, but do not need to simplify your final answer, as long as it is in a form where it could be entered into a calculator.

(c) Do you think your approximation is an overestimate or an underestimate of the true answer? Explain your reasoning.

  1. (20 pts. – 5 pts. each) Answer in a few sentences. You may draw a picture if that helps your explanation, but you must use words as well.

(a) Under what conditions can you be sure a midpoint sum approximation for a definite integral will be smaller than the true value? Explain.

(b) What is the main idea behind Simpson’s approximation for definite integrals? Why does it seem reasonable that this idea should result in a more accurate approximation than is produced by any of the other four approximation methods you know?

(c) Increasing N (the number of divisions used) in any of the methods of approximating integrals generally leads to more accurate approximations. How do you expect increasing N by a factor of 10 to affect the error in a left sum? a trapezoid sum? a Simpson’s sum?

(d) Thinking graphically, it seems natural that

0

sin x dx would be zero, since areas above and below the x-axis should cancel. However, this reasoning is wrong. Why? Explain.