MA 141 Lecture 23: Fundamental Theorem of CalculusExam
Course CodeMA 141
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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
for all x∈[a, b]is an antiderivative of fon [a, b].
Part 2: If Fis any antiderivative of fon [a, b], then
Part 2 is fundamental to our computation of deﬁnite integrals as we develop more tools. A
common notation will be
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
f(t)dt =F(1) −F(a)
A(−2) = Z−2
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
Part 1 of the FTOC is often used to give the derivative of an integral.
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