Consider the integral ∫ 0 ∞ 4/x^2 - 16 dx a) Rewrite this integral as the combination of improper integrals of Type 1 and Type 2 necessary to evaluate the convergence of the integral. But do not evaluate them. b) Evaluate the indefinite integral ∫ 0 ∞ 4/x^2 - 16 dx c) Determine whether the integral ∫ 0 ∞ 4/x^2 - 16 dx converges or diverges. d) It is only necessary to evaluate one of the Type 1 or Type 2 indefinite integrals from part (a) to determine the result in part (c). Indicate which one and explain why.
Use the Comparison Test for Improper Integrals to determine whether the following integral converges or diverge. ∫ 1 ∞ |cos x|/x^2 + 3x + 4 dx
Consider the arc length of the curve x = ln y on the interval 1 ≤ y ≤ 2. a) Set up and simplify, but do not solve, the integral for the exact length of this curve with respect to y. b) Use a trigonometric substitution to find the exact arc length.
Consider the portion of a hyperbola y = √x^2 - 1, on √2 ≤ x ≤ 2. a) Set up, but DO NOT evaluate, the integral with respect to x that represents the surface area obtained by rotating the curve about the y-axis. b) Set up AND EVALUATE the integral with respect to y which represents the area of the surface obtained by rating the curve about the y-axis.