12. Techniques of Integration

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀² (s + 1) / √(4 − s²) ds

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

b. Show that the inner and outer surfaces of the solid have infinite area.

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")

d. Prove this property holds for any a ≥ 0 and c > 0:

The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass

(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ dr / r^0.999

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to ∞ of (dθ / (1 + e^θ))

Which of the following integrals is improper?

89–91. Comparison Test Determine whether the following integrals converge or diverge.

89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

a. Find A(a,b), the area of R as a function of a and b.

Evaluate the improper integrals in Exercises 53–62.

∫ from 0 to 2 of (1 / (y − 1)^(2/3)) dy

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

13. ∫ (from 0 to ∞) cos x dx

Finding volume

The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

86. ∫ (from -∞ to ∞) x³/(1 + x⁸) dx

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.

84. ∫ (from 0 to π) sec²x dx*(Note: Potential improperness at x = π/2)*