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**preview**shows page 1. to view the full**4 pages of the document.**MA 141 Chapter 4

Section 4.3: Fundamental Theorem of Calculus

Just like with derivatives, once we have a formal deﬁnition of a deﬁnite integral, we can

develop tools that let us compute the deﬁnite integral in a faster or easier manner.

Fundamental Theorem of Calculus: Let fbe continuous on [a, b].

Part 1: The function Adeﬁned by

A(x) = Zx

a

f(t)dt

for all x∈[a, b]is an antiderivative of fon [a, b].

Part 2: If Fis any antiderivative of fon [a, b], then

Zb

a

f(x)dx =F(b)−F(a).

Part 2 is fundamental to our computation of deﬁnite integrals as we develop more tools. A

common notation will be

Zb

a

f(x)dx =F(x)

b

a=F(b)−F(a).

What does Part 1 say? First, Part 1 says that the deﬁnite integral of ffrom ato some

endpoint x, or the area under the curve on [a, x]as xvaries, can be calculated using an

antiderivative of f.

Combining Parts 1 and 2, we get a better idea what this means. Let Fbe any antiderivative

of f. Then

A(1) = Z1

a

f(t)dt =F(1) −F(a)

A(−2) = Z−2

a

f(t)dt =F(−2) −F(a)

In addition, this is equivalent to saying

A(x) = F(x)−F(a)

A′(x) = F′(x)−0

=f(x)

Part 1 of the FTOC is often used to give the derivative of an integral.

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