Shortcut Methods
Numerical of Indefinite Integral
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Power Rule:
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Sum Rule:
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Difference Rule:
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Constant Multiple Rule:
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Substitution Rule: If (u = g(x)) is a differentiable function, then
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Logarithmic rule:
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Exponential Rule:
Numerical of Definite Integral
where (F(x)) is an antiderivative of (f(x)) (i.e. (\frac{d}{dx}F(x) = f(x))).
Applications of Integral Calculus
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Area under a curve:
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Volume of a solid generated by revolving a region around an axis:
where (A(x)) is the area of the cross-section of the solid at (x).
- Work done by a force:
where (F(x)) is the force applied at the point (x).
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Average value of a function:
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Probability:
where (f(x)) is the probability density function of the random variable (X).
