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Problems in Calculus: Ordering Definite Integrals and Symmetries, Assignments of Analytical Geometry and Calculus

Western Washington University (WWU)Analytical Geometry and Calculus

Two calculus problems related to section 5.4. The first problem deals with the function f(x) and its definite integrals i1, i2, i3, ..., and asks to arrange them in increasing order using properties of definite integrals. The second problem involves the function g(x) = (sin x)2 and calculating the exact values of its definite integrals from 0 to π and 0 to π/2. Both problems require a deep understanding of calculus concepts such as definite integrals, properties of integrals, and symmetries.

Typology: Assignments

Pre 2010

Uploaded on 08/16/2009

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koofers-user-mjr 🇺🇸

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Two problems related to Section 5.4
Problem 1. Consider the function
f(x) =
sin x
xfor x6= 0
1 for x= 0.
As it was discussed in Math 124, this function is continuous. Let nbe a positive
integer and consider the definite integrals
In=Znπ
0
f(x)dx, n = 1,2,3, . . . .
(a) Use properties of the definite integral discussed in Section 5.4 to arrange the
numbers I1, I2, I3, I4, I5, . . . in increasing order.
(b) Explain your reasoning by stating explicitly which properties you use and how
they apply to the definite integrals I1, I2, I3, I4, I5, . . ..
(c) Do you recognize a pattern in the ordering of the numbers
I1, I2, I3, I4, I5, I6, I7, I8, I9, I10, I11, . . . ?
State this pattern clearly.
Problem 2. Consider the function g(x) = (sin x)2.
(a) The function ghas symmetries which can help you calculate the definite integrals
below. Discover these symmetries and explain them.
(b) Calculate the exact value of Zπ
0
g(x)dx.
(c) Calculate the exact value of Zπ/2
0
g(x)dx.
Give detailed explanations of your reasoning.
1

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Two problems related to Section 5.

Problem 1. Consider the function

f (x) =

sin x x

for x 6 = 0

1 for x = 0.

As it was discussed in Math 124, this function is continuous. Let n be a positive integer and consider the definite integrals

In =

∫ (^) nπ

0

f (x) dx, n = 1, 2 , 3 ,....

(a) Use properties of the definite integral discussed in Section 5.4 to arrange the numbers I 1 , I 2 , I 3 , I 4 , I 5 ,... in increasing order.

(b) Explain your reasoning by stating explicitly which properties you use and how they apply to the definite integrals I 1 , I 2 , I 3 , I 4 , I 5 ,.. ..

(c) Do you recognize a pattern in the ordering of the numbers

I 1 , I 2 , I 3 , I 4 , I 5 , I 6 , I 7 , I 8 , I 9 , I 10 , I 11 ,...?

State this pattern clearly.

Problem 2. Consider the function g(x) = (sin x)^2.

(a) The function g has symmetries which can help you calculate the definite integrals below. Discover these symmetries and explain them.

(b) Calculate the exact value of

∫ (^) π

0

g(x) dx.

(c) Calculate the exact value of

∫ (^) π/ 2

0

g(x) dx.

Give detailed explanations of your reasoning.