69-72. Volumes of solids Find the volume of the following solids.
70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.

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Identify the region bounded by the curves: the function \(y = \frac{1}{x^{2}(x^{2} + 2)^{2}}\), the line \(y = 0\), and the vertical lines \(x = 1\) and \(x = 2\).

Since the solid is formed by revolving the region about the y-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells is: \(V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\).

In this problem, the radius of a shell is the distance from the y-axis to the shell, which is \(x\), and the height of the shell is the function value \(y = \frac{1}{x^{2}(x^{2} + 2)^{2}}\).

Set up the integral for the volume as: \(V = \int_{1}^{2} 2\pi x \cdot \frac{1}{x^{2}(x^{2} + 2)^{2}} \, dx\).

Simplify the integrand before integrating, then evaluate the integral to find the volume of the solid.

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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 can be computed using methods like the disk/washer or shell method, depending on the axis of rotation and the shape of the region.

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Finding Volume Using Disks

Shell Method

The shell method calculates volume by integrating cylindrical shells formed when a region is revolved around an axis. It is especially useful when revolving around the y-axis and involves integrating 2π(radius)(height) with respect to x or y.

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Euler's Method

Setting up Proper Integral Limits and Functions

Accurately identifying the bounds of integration and the function expressions for radius and height is crucial. For the given problem, the limits are x=1 to x=2, and the function y = 1/[x²(x² + 2)²] defines the height of the shell or the radius depending on the method.

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