Question

Finding volumeThe infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

Accepted Answer

Identify the region bounded by the coordinate axes and the curve $$y = -\ln x in $$the first quadrant. Since $$y = -\ln x$$, and we are in the first quadrant, $$x$$ ranges from 0 to 1 because $$\ln x is $$negative for $$0 \u003c x \u003c 1$$ and $$y is $$positive there. Set up the volume integral using the method of disks/washers. When revolving around the x-axis, the volume element is $$dV = \pi [f(x)]^2 dx$$, where $$f(x) is $$the radius of the disk. Here, the radius is $$y = -\ln x. $$Write the volume integral as $$V = \pi \$$int_0$$^1 (-\ln x)^2$ \, dx. $$This integral represents the volume of the solid generated by revolving the region around the x-axis. To evaluate the integral $$\$$int_0$$^1 (-\ln x)^2$ \, dx$$, consider using the substitution $$t = -\ln x$$, which implies $$x = e$$^{-t}$$ and dx$$ = $$-e^{-t} dt$. Change the limits accordingly: when x=0$$^+$$, t$$ 	o \infty$$, and when x=1$, t=0$. Rewrite the integral in terms of t$ and evaluate it using integration techniques for exponential and polynomial functions. This will give the volume of the solid.

Finding volume
The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

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Key Concepts

Volume of Solids of Revolution

Natural Logarithm Function and Its Properties

Definite Integration and Limits of Integration