Domanda del libro di testo
[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.3.73
Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.
Guida verificata passo dopo passo1
Identify the interval for one arch of the curve \( y = \sin x \). Since one arch corresponds to one complete wave from 0 to \( \pi \), set the limits of integration as \( x = 0 \) to \( x = \pi \).
Recall the formula for the volume \( V \) generated by revolving a curve \( y = f(x) \) about the x-axis from \( x = a \) to \( x = b \):
\[ V = \pi \int_{a}^{b} [f(x)]^{2} \, dx \]
Substitute \( f(x) = \sin x \) and the limits \( a = 0 \), \( b = \pi \) into the volume formula:
\[ V = \pi \int_{0}^{\pi} (\sin x)^{2} \, dx \]
Use the trigonometric identity to simplify \( (\sin x)^2 \):
\[ \sin^{2} x = \frac{1 - \cos(2x)}{2} \]
Rewrite the integral using this identity.
Set up the integral for evaluation:
\[ V = \pi \int_{0}^{\pi} \frac{1 - \cos(2x)}{2} \, dx \]
From here, you can integrate term-by-term to find the volume.
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Volume of Solids of Revolution
This concept involves finding the volume of a 3D solid formed by rotating a 2D curve around an axis. The volume is typically calculated using integral calculus, where the shape is sliced into thin disks or washers perpendicular to the axis of rotation.
Video consigliato:
04:48
Finding Volume Using Disks
Disk Method
The disk method calculates volume by summing up the volumes of infinitesimally thin circular disks formed when a region is revolved around an axis. Each disk's volume is π(radius)^2 times the thickness, and integration over the interval gives the total volume.
Video consigliato:
06:30Disk Method Using y-Axis
Properties of the Sine Function
Understanding the sine function, especially one arch from 0 to π, is crucial. The function y = sin x is positive and continuous in this interval, which defines the shape of the region being revolved and sets the limits of integration for the volume calculation.
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Percorso guidato
06:21Properties of Functions
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