Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.

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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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주요 개념

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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.

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.

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.