how is it useful in our lives? any real-world applications?
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Since you ask about the utility, did you try searching on that question already? Here are two such results:
https://www.toppr.com/bytes/calculus-in-everyday-life/
https://www.quora.com/What-are-some-real-life-applications-of-integration-and-differentiation
Places where integration occurs in settings relevant to the real world include probability (continuous probability distributions like the bell curve or normal distribution are described in terms of an integral), medical imaging technology (the math behind CT scans uses integrals such as the Radon transform), and Fourier transforms (used in the analysis of signals of all kinds).
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