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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.2.44
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.

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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:
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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:
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Euler'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.
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Properties of Functions
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