Fundamental Theorem of Calculus

integrate

π

4

0

net area = 0.

blue  positive area red  negative area

The fundamental theorem of calculus states that if

f

is continuous on

[a,b]

, then the function

g

defined on

[a,b]

by

g(x)=

x

∫

a

f(t)dt

is continuous on

[a,b]

, differentiable on

(a,b)

, and

g'(x)=f(x)

. This Demonstration illustrates the theorem using the cosine function for

f(x)

. As you drag the slider from left to right, the net area between the curve and the

x

axis is calculated and shown in the upper plot, with the positive signed area (above the

x

axis) in blue and negative signed area (below the

x

axis) in red. The lower plot shows the resulting area values versus position

x

.