Calculus2

Improper Integrals

� 1, 2 For the given area function A(t), do the following:

a) Compute the integral to find the formula for A(t).

b) Use your formula to evaluate A(t) at t = 10, 100, and 1000.

c) Calculate lim t→∞

A(t).

1. A(t) =

∫ t 1

1

x3 dx.

2. A(t) =

∫ t 1

1 √ x dx.

� 3–10 (a) Sketch the curve at issue, and (b) compute the integral.

3.

∫ ∞ 0

x

(x2 + 1)2 dx 4.

∫ ∞ 2π

sin θdθ

5.

∫ ∞ 1

ln x

x dx 6.

∫ ∞ 1

ln x

x2 dx

7.

∫ ∞ −∞

ye−y 2

dy 8.

∫ 3 2

1 √

3 −z dz

9.

∫ 1 −1

ex

ex − 1 dx 10.

∫ 1 0

1 √

1 −x2 dx Hint: You may use tables.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11. Consider

∫ ∞ 1

1

xp dx.

We showed in class that this integral diverges when p = 1. Determine whether the integral converges or diverges for (a) p > 1 and (b) p < 1.

Hint: You only need to compute the integral once. Then evaluate the resulting limit for the cases separately.

� 12, 13 Use the Comparison Test to determine whether the integral is convergent or divergent. You may cite your result from Problem (11) if applicable.

12.

∫ ∞ 1

x √ x3 − 1

dx

13.

∫ ∞ 1

cos2 x

1 + x2 dx Hint: cos2 x ≤ 1 for all x.

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Solutions to Selected Problems

1. a) A(t) = t2 − 1

2t2

b) A(10) = 0.495 A(100) = 0.49995 A(1, 000) = 0.4999995

c) 1 2

2. a) A(t) = 2 √ t− 2

b) A(10) ≈ 4.32 A(100) = 18 A(1, 000) ≈ 61.25

c) ∞

3. 1 2

4. Diverges

5. Diverges

6. 1

7. 0

8. 2

9. Diverges

10. π 2

11. a) Converges if p > 1

b) Diverges if p < 1

12. Divergent

13. Convergent

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