Below are statements of theorems from integral calculus with accompanying visualizations.

If [math]f[/math] is continuous on [math][a,b][/math], then:[br][list=1][*]If [math]g\left(x\right)=\int_a^xf\left(t\right)dt[/math], then [math]g'\left(x\right)=f\left(x\right)[/math]. Another way of saying this is [math]\frac{d}{dx}\int_a^xf(t)dt=f(x)[/math].[/*][*][math]\int_a^bf(x)dx=F(b)-F(a)[/math], where [math]F[/math] is any antiderivative of [math]f[/math] (that is, [math]F'=f[/math]).[/*][/list]

If [math]f[/math] is continuous on [math][a,b][/math], then there is a number [math]c[/math] in [math](a,b)[/math] such that [math]f(c)=\frac{1}{b-a}\int_a^bf(x)dx[/math].