Volumes
Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 about
a. the x-axis

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Identify the region bounded by the curves: the x-axis (y = 0), the curve y = 3x^{4}, and the vertical lines x = -1 and x = 1. This region lies between x = -1 and x = 1 and above the x-axis.

Since the solid is generated by revolving the region about the x-axis, use the disk method to find the volume. The volume of a solid of revolution about the x-axis is given by the integral: \[ V = \pi \int_{a}^{b} [f(x)]^{2} \, dx \] where $f(x)$ is the function representing the curve, and $a$ and $b$ are the bounds of integration.

Substitute the function and bounds into the formula. Here, $f(x) = 3x^{4}$, $a = -1$, and $b = 1$, so the volume integral becomes: \[ V = \pi \int_{-1}^{1} (3x^{4})^{2} \, dx \]

Simplify the integrand inside the integral: \[ (3x^{4})^{2} = 9x^{8} \] So the integral becomes: \[ V = \pi \int_{-1}^{1} 9x^{8} \, dx \]

Evaluate the integral by integrating \$9x^{8}$ with respect to $x$ over the interval $[-1, 1]$. After finding the antiderivative, apply the Fundamental Theorem of Calculus to compute the definite integral, then multiply by $\pi$ 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 region 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 the region is revolved around an axis. Each disk's volume is π(radius)^2 times its thickness, and the radius is given by the function value at each x.

Setting up Definite Integrals with Boundaries

To find the volume, you must correctly identify the limits of integration and the function defining the radius. Here, the region is bounded by x = -1 and x = 1, and the radius is given by y = 3x^4, which determines the size of each disk when revolving around the x-axis.

VolumesFind the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 abouta. the x-axis