A line integral is a simple integral.
they look like:
integral x=a to b of (f(x)).
A surface integral is an integral of two variables.
they look like:
integral x=a to b, y=c to d of (f(x,y)).
or
integral x=a to b of (integral y=c to d of (f(x,y))).
The second form is the nested form. A pair of line integrals,
one inside the other. This is the easiest way to understand surface
integrals, and, normally, solve surface integrals.
A volume integral is an integral of three variables.
they look like:
integral x=a to b, y=c to d, z=e to f of (f(x,y,z)).
or
integral x=a to b of (integral y=c to d of (integral z=e to f of
f(x,y,z))).
the above statement is wrong, the person who wrote this stated
the first 2 types of integrals as regular, simple, scalar
integrals, when line and surface integrals are actually a form of
vector calculus. in the previous answer, it is stated that the
integrand is just some funtion of x when it is actually usually a
vector field and instead of evaluating the integral from some x a
to b, you will actually be evaluating the integral along a curve
that you will parametrize to get the upper and lower bounds of the
integral. as you can see, these are a lot more complicated.
looking at your question tho, i dont think you want the whole
expanation on how to solve these problems, but more so what they
are and what they are used for, because these can be a pain to
solve and there are also several ways to solve them indirectly.
line integrals have an important part in physics because they
alow us to calculate things such as work that have vector values
rather than just scalar values as you can use these integrals to
describe a particles path along a curve in a force field. surface
integrals help us calculate things like flux, or how fluid flows
over a surface.
if you want to learn more, look into things like greens theorem,
or the divergence theorem.
p.s. his definition of a surface integral is acutally how you
find the volume of a region
