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when i was in highschool taking calculus, a fair bit of our textbook was devoted to deriving the basic equations for velocity and other rates of change. there was also a section on water passing through cones or cylinders. say you have an icecream cone filled with water, and you cut out the bottom, how can you determine the water level after it has dripped out for a certain amount of time. these are simple examples of differential calculus in action.
integrals are all about areas and volumes in the introductory courses.
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