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
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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 8.2.44
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.
Verified step by step guidance1
Identify the region bounded by the curve \(f(x) = \sin(x)\) and the x-axis on the interval \([0, \pi]\). This region lies between the curve and the x-axis from \(x=0\) to \(x=\pi\).
Since the solid is generated by revolving this region about the y-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells is:
\(V = \int_a^b 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\)
In this problem, the radius of a shell at position \(x\) is the distance from the y-axis, which is \(x\), and the height of the shell is the function value \(\sin(x)\). So the volume integral becomes:
\(V = \int_0^{\pi} 2\pi x \sin(x) \, dx\)
Set up the integral \(\int_0^{\pi} 2\pi x \sin(x) \, dx\) and prepare to evaluate it. You can factor out the constant \(2\pi\) to simplify the integral to \(2\pi \int_0^{\pi} x \sin(x) \, dx\).
To evaluate \(\int_0^{\pi} x \sin(x) \, dx\), use integration by parts where you let \(u = x\) (so \(du = dx\)) and \(dv = \sin(x) dx\) (so \(v = -\cos(x)\)). Apply the integration by parts formula:
\(\int u \, dv = uv - \int v \, du\).
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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. The volume is typically calculated using integral methods such as the disk, washer, or shell method, depending on the axis of rotation and the shape of the region.
Recommended video:
04:48
Finding Volume Using Disks
Shell Method
The shell method is used to find volumes when revolving a region around a vertical or horizontal axis, especially when the axis is parallel to the axis of the function. It involves integrating cylindrical shells, where the volume element is 2π(radius)(height)(thickness).
Recommended video:
07:33Euler's Method
Properties of the Sine Function on [0, π]
Understanding the behavior of f(x) = sin(x) on the interval [0, π] is crucial, as it is positive and continuous, forming a single 'hump' above the x-axis. This ensures the region bounded by sin(x) and the x-axis is well-defined and suitable for volume calculations.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
7–84. Evaluate the following integrals.
11. ∫ from 0 to π/4 (sec x – cos x)² dx
Textbook Question
7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx
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?
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Textbook Question
69-72. Volumes of solids Find the volume of the following solids.
70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.
Textbook Question
7–84. Evaluate the following integrals.
25. ∫ [1 / (x√(1 - x²))] dx