# Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

The fundamental theorem of calculus states that if is continuous on , then the function defined on by is continuous on , differentiable on , and . This Demonstration illustrates the theorem using the cosine function for . As you drag the slider from left to right, the net area between the curve and the axis is calculated and shown in the upper plot, with the positive signed area (above the axis) in blue and negative signed area (below the axis) in red. The lower plot shows the resulting area values versus position .

f

[a,b]

g

[a,b]

g(x)=f(t)dt

x

∫

a

[a,b]

(a,b)

g'(x)=f(x)

f(x)

x

x

x

x