MHB Sparkling's question at Yahoo Answers regarding a volume by slicing

[MarkFL]

Here is the question:

How to find the volume of a solid given an equation and bounds?

Let the first quadrant region enclosed by the graph of =1/x, x=1 and x=4 be the base of a solid. If cross sections perpendicular to the x-axis are semicircles, the volume of the solid is:

(Answer: 3pi/ 32)

Please tell me how you got the answer because I kept getting 3pi/4

I have posted a link there to this thread so the OP can view my work.

 

[MarkFL]

Hello Sparkling,

The volume of an arbitrary semicircular slice is:

$$dV=\frac{\pi}{8}D^2\,dx$$

where the diameter $D$ is:

$$D=\frac{1}{x}$$

hence:

$$dV=\frac{\pi}{8}x^{-2}\,dx$$

And so, the sum of all the slices is given by:

$$V=\frac{\pi}{8}\int_1^4 x^{-2}\,dx$$

Applying the FTOC, we obtain:

$$V=\frac{\pi}{8}\left[-\frac{1}{x} \right]_1^4=\frac{\pi}{8}\left(1-\frac{1}{4} \right)=\frac{3\pi}{32}$$