The Fundamental Theorem of Calculus

Formally speaking, the fundamental theorem of calculus is split into two parts.

Loosely speaking, and for purposes here, we may say that the fundamental theorem of calculus states that

\int_a^b f(x) \, dx=F(b)-F(a)

where F(x) is the antiderivative of f(x), that is to say


Example – calculate \int_1^4 2x^3\, dx.

If f(x)=2x^3, then the antiderivative is given by


see Integrating Polynomials. It follows from the Fundamental Theorem of Calculus that

\int_4^1 2x^3\, dx=F(4)-F(1)=\frac{1}{2}(4)^4-\frac{1}{2}(1)^4=128-\frac{1}{4}=127\frac{3}{4}