custom directory for rstudio server on droplet

I have a website that is currently working (say, example.com). I have set up my virtual host file (example.conf) so that port 80 points to example.com:

<VirtualHost *:80>

	ServerAdmin admin@example.com
        ServerName example.com
        ServerAlias www.example.com
	DocumentRoot /var/www/example

	ErrorLog ${APACHE_LOG_DIR}/error.log
	CustomLog ${APACHE_LOG_DIR}/access.log combined


RewriteEngine on
RewriteCond %{SERVER_NAME} =www.example.com [OR]
RewriteCond %{SERVER_NAME} =example.com
RewriteRule ^ https://%{SERVER_NAME}%{REQUEST_URI} [END,NE,R=permanent]

</VirtualHost>

I have also set up ssl in a file called example-le-ssl.conf:

<IfModule mod_ssl.c>
<VirtualHost *:443>

	ServerAdmin admin@example.com
        ServerName example.com
        ServerAlias www.example.com
	DocumentRoot /var/www/example

	ErrorLog ${APACHE_LOG_DIR}/error.log
	CustomLog ${APACHE_LOG_DIR}/access.log combined

Include /etc/letsencrypt/options-ssl-apache.conf
SSLCertificateFile /etc/letsencrypt/live/example.com/fullchain.pem
SSLCertificateKeyFile /etc/letsencrypt/live/example.com/privkey.pem
</VirtualHost>
</IfModule>

So far, so could. Now, I have installed a shiny server (which defaults to port 3838). I can access that by typing example.com:3838, but I want to access it by typing example.com/shiny. I read the instructions on how to set up a reverse proxy. In there, it says to replace 000-default.conf with:

<VirtualHost *:80>
    ProxyPreserveHost On

    ProxyPass / http://127.0.0.1:8080/
    ProxyPassReverse / http://127.0.0.1:8080/
</VirtualHost>

Of course, in my case, I assume I would do 3838 instead of 8080:

<VirtualHost *:80>
    ProxyPreserveHost On

    ProxyPass /shiny/ http://127.0.0.1:3838/
    ProxyPassReverse /shiny http://127.0.0.1:3838/
</VirtualHost>

No luck. example.com/shiny can’t be found.

So, I change the ip address to my site’s url:

<VirtualHost *:80>
    ProxyPreserveHost On

    ProxyPass /shiny/ http://example.com:3838/
    ProxyPassReverse /shiny http://example.com:3838/
</VirtualHost>

Still no luck, so I try my ip address:

<VirtualHost *:80>
    ProxyPreserveHost On

    ProxyPass /shiny/ http://[some ip address]:3838/
    ProxyPassReverse /shiny http://[some ip address]:3838/
</VirtualHost>

Still no luck. So I think that maybe I need to change example.conf instead of 000-default.conf:

<VirtualHost *:80>

	ServerAdmin admin@example.com
        ServerName example.com
        ServerAlias www.example.com
	DocumentRoot /var/www/example

	ErrorLog ${APACHE_LOG_DIR}/error.log
	CustomLog ${APACHE_LOG_DIR}/access.log combined


RewriteEngine on
RewriteCond %{SERVER_NAME} =www.example.com [OR]
RewriteCond %{SERVER_NAME} =example.com
RewriteRule ^ https://%{SERVER_NAME}%{REQUEST_URI} [END,NE,R=permanent]

**    ProxyPreserveHost On

    ProxyPass /shiny/ http://[some ip address]:3838/
    ProxyPassReverse /shiny http://[some ip address]:3838/**

</VirtualHost>
```

Here I also tried the ip address, 127.0.0.1, and example.com. No luck.

What am I missing?