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Final

MA 141 Lecture 23: Fundamental Theorem of CalculusExam


Department
Mathematics
Course Code
MA 141
Professor
Kevin Flores
Study Guide
Final

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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
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) = Z2
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.
1
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