TITLE: Solving $\sin(x) = \ln(x)$
QUESTION [7 upvotes]: While tutoring a student in introductory calculus, I inquired as to how she would set up an integral to determine the volume of the solid created by revolving the region bounded by the functions $\sin(x)$, $\ln(x)$, and the $x$-axis about the line $y=5$. She set up the integral as follows: $$\pi\int_{1}^{a} 25-(\ln(x))^2 dx + \pi\int_{a}^{\pi} 25-\sin^2(x)dx $$ where $a$ is the solution to $\sin(x) = \ln(x)$. However, I couldn't give an explanation as to how to find this point. Is this something that could be solved given two years of introductory calculus, or does it require a larger toolkit? In searching for an answer to this problem online, I was bombarded by hits on $\ln(x)$ differentiation resources which obviously weren't helpful. Any light you could shed would be appreciated!

REPLY [4 votes]: Numerical methods are the way to go. Clearly there would be only one solution which can be figured by considering the monotonicity of $\ln x$ and the boundedness of $\sin x$. It would help if you're allowed to use a calculator.
Define $f(x)=\sin x-\ln x$. By Newton Raphson method we have the following iterative formula.
$$x_{n+1}=x_n-\dfrac{\sin x_n-\ln x_n}{\cos x_n-1/x_n}$$
I'm going to make a first guess say $x_0=2$. Now use the iteration formula to obtain an accurate value of the solution: $$\begin{array}{|p{3cm}||p{3cm}|p{3cm}|p{3cm}|}\hline n & x_n  \\ \hline 0 & 2 \\ 1 & 2.23593406389 \\ 2& 2.21918552153 \\ 3& 2.21910715064\\\hline \end{array}$$