[PDF] Math 104: Improper Integrals (With Solutions)





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  • Comment expliquer la convergence ?

    ? convergence

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  • Si une série est convergente, alors S = Sn + Rn (pour tout n ? 0) et limn?+? Rn = 0. uk = Sn + Rn. Donc Rn = S ? Sn ? S ? S = 0 lorsque n ? +?.
Math 104: Improper Integrals (With Solutions)

Math 104: Improper Integrals (With Solutions)

Ryan Blair

University of Pennsylvania

Tuesday March 12, 2013

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 1 / 15

Outline

1Improper Integrals

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 2 / 15

Improper Integrals

Improper integrals

Definite integrals?

b a f(x)dxwere required to have finite domain of integration [a,b] finite integrandf(x)<±∞ Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 3 / 15

Improper Integrals

Improper integrals

Definite integrals?

b a f(x)dxwere required to have finite domain of integration [a,b] finite integrandf(x)<±∞

Improper integrals

1Infinite limits of integration

2Integrals with vertical asymptotes i.e. with infinite discontinuity

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 3 / 15

Improper Integrals

Infinite limits of integration

Definition

Improper integrals are said to be

convergentif the limit is finite and that limit is the value of the improper integral. divergentif the limit does not exist. Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 4 / 15

Improper Integrals

Infinite limits of integration

Definition

Improper integrals are said to be

convergentif the limit is finite and that limit is the value of the improper integral. divergentif the limit does not exist. Each integral on the previous page is defined as a limit. If the limit is finite we say the integralconverges, while if the limit is infinite or does not exist, we say the integraldiverges. Convergence is good (means we can do the integral); divergence is bad (means we can"t do the integral). Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 4 / 15

Improper Integrals

Example 1

Find?∞

0 e-xdx. (if it even converges)

Solution:

0 e-xdx = limb→∞? b 0 e-xdx= limb→∞? -e-x?b0 = lim b→∞-e-b+e0= 0 + 1 = 1.

So the integral converges and equals 1.

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 5 / 15

Improper Integrals

Example 2

Find?∞

-∞1

1 +x2dx.

(if it even converges)

Solution: By definition,

-∞1

1 +x2dx=?

c -∞11 +x2dx+? c11 +x2dx, where we get to pick whatevercwe want. Let"s pickc= 0. ?0 -∞1

1 +x2dx= limb→-∞?

arctan(x)? 0 b= limb→-∞[arctan(0)-arctan(b)] = 0-limb→-∞arctan(b) =π2 Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 6 / 15

Improper Integrals

Example 2, continued

Similarly,?∞

01

1 +x2dx=π2.

Therefore,

-∞1

1 +x2dx=π2+π2=π.

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 7 / 15

Improper Integrals

Example 3, thep-test

The integral?∞

11 xpdx

1Convergesifp>1;

For example:

11 x3/2dx= limb→∞-?2x1/2? b 1= 2, while 11 x1/2dx= limb→∞?

2⎷x?

b

1= limb→∞2⎷b-2=∞,

and 11 xdx= limb→∞? ln(x)? b

1= limb→∞ln(b)-0=∞.

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 8 / 15

Improper Integrals

Convergence vs. Divergence

In each case, if the limit exists (or if both limits exist, in case 3!), we say the improper integralconverges. If the limit fails to exist or is infinite, the integraldiverges. In case 3, if eitherlimit fails to exist or is infinite, the integral diverges. Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 9 / 15

Improper Integrals

Example 4

Find?2

02x x2-4dx. (if it converges)

Solution: The denominator of2x

x2-4is 0 whenx= 2, so the function is not even defined whenx= 2. So ?2 02x x2-4dx= limc→2-? c

02xx2-4dx= limc→2-?

ln|x2-4|? c 0 = lim c→2-ln|x2-4| -ln(4) so the integral diverges. Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 10 / 15

Improper Integrals

Example 5

Find? 3

01(x-1)2/3dx,if it converges.

Solution:

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 11 / 15

Improper Integrals

Example 5

Find? 3

01(x-1)2/3dx,if it converges.

Solution: We might think just to do

3 01 (x-1)2/3dx=?

3(x-1)1/3?30,

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 11 / 15

Improper Integrals

Example 5

Find? 3

01(x-1)2/3dx,if it converges.

Solution: We might think just to do

3 01 (x-1)2/3dx=?

3(x-1)1/3?30,

but this is not okay: The functionf(x) =1 (x-1)2/3isundefined when x= 1 , so we need to split the problem into two integrals. 3 01 (x-1)2/3dx=? 1

01(x-1)2/3dx+?

3

11(x-1)2/3dx.

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 11 / 15

Improper Integrals

Example 5

Find? 3

01(x-1)2/3dx,if it converges.

Solution: We might think just to do

3 01 (x-1)2/3dx=?

3(x-1)1/3?30,

but this is not okay: The functionf(x) =1 (x-1)2/3isundefined when x= 1 , so we need to split the problem into two integrals. 3 01 (x-1)2/3dx=? 1

01(x-1)2/3dx+?

3

11(x-1)2/3dx.

The two integrals on the right hand side both converge and addup to

3[1 + 21/3], so?3

01 (x-1)2/3dx= 3[1 + 21/3]. Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 11 / 15

Improper Integrals

Tests for convergence and divergence

The gist:

1If you"re smaller than something that converges, then youconverge.

2If you"re bigger than something that diverges, then you diverge.

Theorem

x≥a. Then

1?∞

af(x)dx converges if?∞ ag(x)dx converges.

2?∞

ag(x)dx diverges if?∞ af(x)dx diverges. Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 12 / 15

Improper Integrals

Example 6

Which of the following integrals converge?

(a)? 1 e-x2dx,(b)? 1sin 2(x) x2dx.

Solution: Both integrals converge.

see?∞

1e-xdx=1

e, so?∞

1e-x2dxconverges.

for allx≥1. Since?∞ 11 x2dxconverges (byp-test), so does?∞ 1sin 2(x) x2dx. Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 13 / 15

Improper Integrals

Limit Comparison Test

Theorem

If positive functions f and g are continuous on[a,∞)and lim x→∞f(x) g(x)=L,0BOTH converge or BOTH diverge.

Example 7

: Letf(x) =1⎷x+1; consider?∞ 11 ⎷x+ 1dx.

Does the integral converge or diverge?

Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 14 / 15

Improper Integrals

Example 7, continued

Solution: We note thatflooks a lot likeg(x) =1⎷x, and?∞

1g(x)dxdiverges by thep-test. Furthermore,

lim x→∞f(x) g(x)=⎷ x⎷x+ 1= 1, so the LCT says 11 ⎷x+1dxdiverges. Ryan Blair (U Penn)Math 104: Improper IntegralsTuesday March 12, 2013 15 / 15quotesdbs_dbs30.pdfusesText_36
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