∫
Definite Integral
Undergraduate
Definition
Represents the area between a function and the x-axis, defined as a limit.
Formulas
∈tₐ^b f(x) dx = lim_n → ∈fty ∑_i=1ⁿ f(xᵢ)Δ x
Defined as limit of Riemann sums
∈tₐ^b f(x) dx = F(b) - F(a)
Fundamental Theorem of Calculus
Examples
Example 1
Find ∫₀² x² dx.
∈t₀² x² dx = (8)/(3)
Example 2
Find ∫₀^π sin x dx.
Applications
Geometry
Area under curves
Physics
Displacement, work
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