Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
85. Find the volume of the solid generated by revolving the region about the y-axis.

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Identify the region bounded by the curve \(y = e^{-x}\), the x-axis, and the y-axis in the first quadrant. This region extends from \(x=0\) to \(x=\infty\) and from \(y=0\) to \(y=1\) since \(e^{-0} = 1\) and \(e^{-\infty} = 0\).

Since the solid is generated 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, which is \(x\), and the height of the shell is the function value \(y = e^{-x}\). The limits of integration are from \(x=0\) to \(x=\infty\).

Set up the integral for the volume: \(V = \int_{0}^{\infty} 2\pi x e^{-x} \, dx\)

To evaluate the integral, use integration by parts where you let one part be \(x\) and the other be \(e^{-x}\). After setting up the integral, proceed with integration by parts 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. 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

The shell method calculates volume by integrating cylindrical shells formed by revolving vertical slices of the region around a vertical axis. Each shell's volume is approximated by its circumference times height times thickness. This method is especially useful when revolving around the y-axis and the function is given in terms of x.

Exponential Decay Function y = e^(-x)

The function y = e^(-x) represents exponential decay, approaching zero as x approaches infinity. Understanding its behavior helps determine the bounds of integration and the shape of the region. Since the region is bounded by this curve and the x-axis in the first quadrant, the limits are from x = 0 to infinity.

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.85. Find the volume of the solid generated by revolving the region about the y-axis.