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A standard calculus textbook has a whole chapter on applications of integrals, so that's what this is. There's a few examples of why these applications are important and interesting throughout. Note that differential equations are broken out into a separate section.
\subsection{Area Between Two Curves}
\subsubsection{Why We Care}
\subsection{Stuff With Solids}
\subsubsection{Volume}
\subsubsection{Arc Length and Surface Area}
\subsubsection{Why We Care}
\subsection{Some Other Common Applications}
This lists common applications that come up in classes but in reality applications for integrals are everywhere. It should be fairly obvious that they're in the hard sciences, but they also show up in economics (), social science (), and even humanities oriented subjects like \_ or \_. It makes sense why the physics applications are the ones always mentioned as they're the obvious, but it'd do everyone a service if the others were mentioned more.
\subsubsection{Work}