Gabriel's horn is formed by revolving the curve $y=1/x$ for $x\in [1,\infty)$ about the $x$-axis.
Find the volume inside Gabriel's horn. I have the answer but I can't seem to get it right. Can someone explain please?
Gabriel's horn is formed by revolving the curve $y=1/x$ for $x\in [1,\infty)$ about the $x$-axis.
Find the volume inside Gabriel's horn. I have the answer but I can't seem to get it right. Can someone explain please?
What you should do for finding the result is to evaluate the following integral $\int_1^\infty \pi y^2 dx$ in which $y=\frac{1}{x}$. You can draw a disc as you see in fig below. We build this disk on $x$- axes, so the volume of it is the volume of colored cylinder. What is that volume? It is $\pi r^2 h$. What is $r$ and what is $h$? indeed, $r$ is $y$ and $h$ is $dx$.
$ x \, y = a^2$
starting point $(a,a)$
end point $(\infty, 0)$
$ \int_a^\infty { \pi y^2 \, dx} = \pi a^3 $
which is $ \frac 34 $ th volume of sphere of radius $a.$