What are Line integral Surface integral and Volume integral in simple words?

Answer 1

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