Evaluate the following integrals. If the integral is not convergent, answer “divergent.”
1.
3.
5.
7. Without integrating, determine whether the integral converges or diverges by comparing the function with .
Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.
9.
11.
13.
15.
17.
19.
21.
23.
25.
Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.
27. ; compare with .
Evaluate the integrals. If the integral diverges, answer “diverges.”
29.
31.
33.
35.
37.
Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.
39.
41.
43.
45. Evaluate . (Be careful!) (Express your answer using three decimal places.)
47. Evaluate .
49. Find the area of the region bounded by the curve , the x-axis, and on the left by .
51. Find the area under in the first quadrant.
53. Find the volume of the solid generated by revolving about the y-axis the region under the curve in the first quadrant.
The Laplace transform of a continuous function over the interval is defined by (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F.
55.
57.
59. Use the formula for arc length to show that the circumference of the circle is .
A non-negative function is a probability density function if it satisfies the following definition: . The probability that a random variable x lies between a and b is given by .
61. Find the probability that x is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy.
Candela Citations
- Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-2/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction