Well, you can use antiderivatives to find the volume of a pear or ring donut, by rotating a curve (or 2 for the donut) about a line. You can have problems stating how many french fries are produced at an amusement park and how many are eaten per hour, and figure out the average rate that they are eaten, or the instantaneous rate at a given time (most likely higher around lunch and dinner times rather than when the park first opens) using derivatives.
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Calculus, both differential and integral.
Calculus; by a long shot.
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.
It is certainly used in calculus, just as calculus can be used in trigonometry.
No. Calculus if a field of mathematics.