Its importance is tremendous - it has many different applications. Some of the applications include calculation of area, of volume, moment of inertia, of work, and many more.
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Most complex engineering problems cannot be solved without calculus. Force related problems are a great example - how else would you calculate the force exerted on a particle a specific distance from an electrically charged wire?
the example and solution of integral calculus
If F(x) is a function, and F ‘(x) = f(x), then F(x) is the antiderivative (or indefinite integral) of f(x) It is the cornerstone of integral calculus and is used for areas, volumes, lengths and so much more!
Just about all of calculus is based on differential and integral calculus, including Calculus 1! However, Calculus 1 is more likely to cover differential calculus, with integral calculus soon after. So there really isn't a right answer for this question.
Gottfried Leibniz is called the father of integral calculus.