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Level 623

Advanced Applications of the Integral

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c*∫f(x) dx
∫c*f(x) dx =
∫[f(x) ± g(x)] dx =
∫f(x) dx ± ∫g(x) dx
∫[f(x) − g(x)] dx on interval [a,b]
If f(x) ≥ g(x) on [a,b], then the area between the curves is
Gini Index =
1/2 * ∫[x − L(x)] dx on interval [0,1]
arclength of f(x) on [a,b]
∫√[1 + (f '(x))²] dx on interval [a,b]
f(x)*g(x) − ∫[f '(x)*g(x)] dx
Integration by Parts: ∫[f(x)*g '(x)] dx =
limit of f '(x) / g '(x)
L'Hopital's Rule: If [limit of f(x) / g(x)] has the form 0/0 or ∞/∞, then [limit of f(x) / g(x)] =
True or False: Continuity implies integrability.
True or False: Integrability implies continuity.
True or False: Differentiability implies integrability.
∫ |f(x)| dx on [a,b]
Area between curve of f(x) and x-axis on interval [a,b]
∫ |v(t)| dt on [a,b]
Distance traveled by object traveling at velocity v(t) on time interval [a,b].
1/(b−a) * ∫ f(x) dx on [a,b]
Average value of function f(x) on interval [a,b]
Fundamental Theorem of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
Mean Value Theorem for Integrals
If f(x) is continuous on [a,b],