Volumes
Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 about
a. the x-axis

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First, identify the region bounded by the curves. The left boundary is given by the parabola \(x = y^{2} + 1\) and the right boundary is the vertical line \(x = 5\). Determine the range of \(y\) values where these two curves intersect by solving \(y^{2} + 1 = 5\).

Solve for \(y\) to find the limits of integration: \(y^{2} = 4\), so \(y = -2\) and \(y = 2\). This means the region extends vertically from \(y = -2\) to \(y = 2\).

Since the solid is generated by revolving the region about the x-axis, consider using the method of cylindrical shells or washers. Here, the washer method is appropriate because the axis of rotation is horizontal and the region is described in terms of \(y\).

Express the radius and thickness of a typical washer. The radius of a washer is the distance from the x-axis to a point \(y\), which is \(|y|\). The thickness is \(dy\). The outer radius corresponds to the line \(x=5\) and the inner radius corresponds to the parabola \(x = y^{2} + 1\).

Set up the volume integral using the washer method: \(V = \pi \int_{-2}^{2} \left[ (5)^{2} - (y^{2} + 1)^{2} \right] dy\). This integral represents the volume of the solid formed by revolving the region around the x-axis.

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주요 개념

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Setting up the region bounded by curves

To find the volume of a solid of revolution, first identify the region bounded by the given curves. Here, the region is between the parabola x = y² + 1 and the vertical line x = 5. Understanding the limits of integration and the shape of the region is essential for correctly applying volume formulas.

Method of cylindrical shells or washers/disks

Volumes of solids of revolution can be found using the disk/washer method or the cylindrical shell method. Since the region is bounded in terms of x and y, and the axis of rotation is the x-axis, choosing the appropriate method involves deciding whether to integrate with respect to x or y and how to express the radius and height of slices.

Revolution about the x-axis and volume integral setup

Revolving a region about the x-axis means each cross-section perpendicular to the x-axis forms a disk or washer. The volume is found by integrating the area of these cross-sections along the axis of revolution. Expressing the radius in terms of y and setting correct bounds is crucial for accurate volume calculation.

VolumesFind the volume of the solid generated by revolving the region bounded on the left by the parabola x = y² + 1 and on the right by the line x = 5 abouta. the x-axis