# The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

If is a continuous function on and is an antiderivative for on , then f(t)t=F(b)-F(a). If we take and for convenience, then is the area under the graph of from to and is the derivative (slope) of . In the image above, the purple curve is —you have three choices—and the blue curve is .

f

[a,b]

F

f

[a,b]

b

∫

a

a=0

F(a)=0

F(x)

f

0

x

f(x)

F(x)

f(x)

F(x)