This is an important application of differential calculus. Related rates is nothing but the study of relations among different variables for the ongoing situation and how they are changing with respect to another quantities. Here you can find real life word problems on related rates. Process of solving such problems is explained beautifully!
There is no difference between linear approximation and linearization. These are just two different ways to say the same thing.
Linear approximation is an important application of differential calculus.
Integral has many applications right from finding areas under the curve to finding volume of solids of revolution, finding arc length, surface area, work done etc and many other applications of physics too.
Here you can find beautiful summary of finding volume by three methods (disk, washer and shell) as well as volume by cross sections along with examples. Finding area under the curve is also discussed