Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
16. ∫ x²/(25 + x²)² dx
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.9.66
65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.
Verified step by step guidance1
Identify the function and the interval: The function is given as \(f(x) = (x^2 + 1)^{-\frac{1}{2}}\), and the interval is \([2, \infty)\).
Set up the volume integral using the disk method since the region is revolved about the x-axis. The volume \(V\) is given by the integral \(V = \pi \int_{2}^{\infty} [f(x)]^2 \, dx\).
Substitute the function into the integral: \(V = \pi \int_{2}^{\infty} \left((x^2 + 1)^{-\frac{1}{2}}\right)^2 \, dx = \pi \int_{2}^{\infty} \frac{1}{x^2 + 1} \, dx\).
Evaluate the improper integral by finding the antiderivative of \(\frac{1}{x^2 + 1}\), which is \(\arctan(x)\), and then apply the limits from 2 to infinity.
Determine if the volume converges by calculating the limit \(\lim_{b \to \infty} \pi [\arctan(b) - \arctan(2)]\). If this limit exists and is finite, the volume exists; otherwise, it does not.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D region around an axis. Common methods include the disk/washer method and the shell method, which use integrals to sum infinitesimal volumes. Understanding how to set up these integrals based on the axis of rotation and the given function is essential.
Recommended video:
04:48
Finding Volume Using Disks
Improper Integrals and Convergence
When the region extends to infinity, the volume integral becomes an improper integral. Determining whether the volume exists requires analyzing the convergence of this integral. Techniques include evaluating limits and applying comparison tests to ensure the integral yields a finite value.
Recommended video:
11:11Improper Integrals: Infinite Intervals
Function Behavior and Boundedness
Understanding the behavior of the function f(x) = (x^2 + 1)^(-1/2) on [2, ∞) is crucial. Since it decreases and approaches zero as x approaches infinity, this affects the shape and size of the solid. Recognizing how the function's decay influences the volume helps in setting up and evaluating the integral correctly.
Recommended video:
5:46Graphs of Exponential Functions
Related Practice
Textbook Question
23-64. Integration Evaluate the following integrals.
26. ∫₀¹ [1 / (t² - 9)] dt
Textbook Question
59. Perpetual Annuity
Imagine that today you deposit \(B in a savings account that earns interest at a rate of *p*% per year compounded continuously (see Section 7.2). The goal is to draw an income of \)I per year from the account forever. The amount of money that must be deposited is:
B = I × ∫(from 0 to ∞) e^(-rt) dt
where r = p/100.
Suppose you find an account that earns 12% interest annually, and you wish to have an income from the account of \$5000 per year. How much must you deposit today?
1
views
Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv
1
views
Textbook Question
7–64. Integration review Evaluate the following integrals.
62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx
Textbook Question
69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-axis?
1
views