Nov. 14, 2016 Worksheet 17: De nite integrals, various ways€¦ · Nov. 14, 2016 Worksheet 17: De...

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Nov. 14, 2016 Worksheet 17: Definite integrals, various ways Fall 2016 In this worksheet, we practice integration by evaluating the definite integral Z π/2 0 cos 2 (x) dx in three different ways. We will need the following two important trig identities: cos 2 (x)= 1 2 (1 + cos(2x)); cos 2 (x) + sin 2 (x)=1. 1. Evaluate Z π/2 0 cos 2 (x) dx by rewriting the integrand, using the first of the above trig identities, and then doing the integration directly (using the Fundamental Theorem of Calculus). Express your answer in terms of π.

Transcript of Nov. 14, 2016 Worksheet 17: De nite integrals, various ways€¦ · Nov. 14, 2016 Worksheet 17: De...

Page 1: Nov. 14, 2016 Worksheet 17: De nite integrals, various ways€¦ · Nov. 14, 2016 Worksheet 17: De nite integrals, various ways Fall 2016 2. (a) Show that, if F(x) = x+ sin(x)cos(x)

Nov. 14, 2016 Worksheet 17: Definite integrals, various ways Fall 2016

In this worksheet, we practice integration by evaluating the definite integral∫ π/2

0

cos2(x) dx

in three different ways.We will need the following two important trig identities:

cos2(x) =1

2(1 + cos(2x)); cos2(x) + sin2(x) = 1.

1. Evaluate ∫ π/2

0

cos2(x) dx

by rewriting the integrand, using the first of the above trig identities, and then doingthe integration directly (using the Fundamental Theorem of Calculus). Express youranswer in terms of π.

Page 2: Nov. 14, 2016 Worksheet 17: De nite integrals, various ways€¦ · Nov. 14, 2016 Worksheet 17: De nite integrals, various ways Fall 2016 2. (a) Show that, if F(x) = x+ sin(x)cos(x)

Nov. 14, 2016 Worksheet 17: Definite integrals, various ways Fall 2016

2. (a) Show that, if

F (x) =x+ sin(x) cos(x)

2,

then F ′(x) = cos2(x). (You will need the second of the two trig identities statedabove to do some simplification after you differentiate.)

(b) Use the result of part (a) above, and the Fundamental Theorem of Calculus, toagain evaluate ∫ π/2

0

cos2(x) dx.

Again, express your answer in terms of π.

Page 3: Nov. 14, 2016 Worksheet 17: De nite integrals, various ways€¦ · Nov. 14, 2016 Worksheet 17: De nite integrals, various ways Fall 2016 2. (a) Show that, if F(x) = x+ sin(x)cos(x)

Nov. 14, 2016 Worksheet 17: Definite integrals, various ways Fall 2016

3. (a) Fill in the five blanks: Since cos2(x)+sin2(x) = 1, we have cos2(x) = 1− ,and therefore,

cos(x) = ±

√1− .

But, since cos(x) ≥ 0 on the interval [0, π/2], we need to take thesign here and not the sign, so we conclude that, on this interval,

(∗) cos(x) =

√1− .

(b) Fill in the blank: using equation (∗) above, we find that

∫ π/2

0

cos2(x) dx =

∫ π/2

0

cos(x) · cos(x) dx =

∫ π/2

0

· cos(x) dx.

(c) Substitute u = sin(x) into the result of part (b) above. Then du =dx; also, when x = 0, u = , and when x = π/2, u = .So we find that ∫ π/2

0

cos2(x) dx =

∫ 1

0

du.

Page 4: Nov. 14, 2016 Worksheet 17: De nite integrals, various ways€¦ · Nov. 14, 2016 Worksheet 17: De nite integrals, various ways Fall 2016 2. (a) Show that, if F(x) = x+ sin(x)cos(x)

Nov. 14, 2016 Worksheet 17: Definite integrals, various ways Fall 2016

(d) Below is a sketch of the function f(u) =√

1− u2.

0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

u

f HuL

What is∫ 1

0f(u) du exactly? (Your answer should involve the quantity π.) Hint:

as you can see, the graph of f(u) on [0, 1] describes a quarter circle.

(e) Use your answers to parts (c) and (d) above to again evaluate∫ π/2

0

cos2(x) dx.