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
c. the line x = 1

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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.

Since the solid is generated by revolving the region about the vertical line x = 1, use the method of cylindrical shells, which is suitable for rotation around vertical lines other than the y-axis.

Set up the volume integral using the shell method formula: \[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx \] where the radius is the distance from the shell to the axis of rotation, and the height is the function value.

Determine the radius of a shell at position x: since the axis of rotation is x = 1, the radius is \(|1 - x|\). The height of the shell is the value of the function \(y = 3x^{4}\).

Set the limits of integration from x = -1 to x = 1, and write the integral explicitly as: \[ V = 2\pi \int_{-1}^{1} (1 - x)(3x^{4}) \, dx \] Note that \(1 - x\) is used because for x in [-1,1], \(1 - x\) is positive.

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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 a given axis. Common methods include the disk/washer method and the shell method, which use integration to sum infinitesimal volumes. Choosing the appropriate method depends on the axis of rotation and the shape of the region.

Shell Method for Volumes

The shell method calculates volume by integrating cylindrical shells formed when revolving a region around a vertical or horizontal line. Each shell's volume is approximated by 2π(radius)(height)(thickness). This method is especially useful when rotating around vertical lines like x = 1, where integrating with respect to y or x simplifies the problem.

Setting up Integration Limits and Radius

Accurately determining the limits of integration and the radius of rotation is crucial. For rotation about x = 1, the radius is the horizontal distance from a point x to the line x = 1, given by |1 - x|. The limits come from the given bounds on x, here from -1 to 1, defining the region to be revolved.

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 aboutc. the line x = 1