AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Functions Defined by Integrals
Ray Cannon
Baylor University
Waco, Texas
While students understand the part of the Fundamental Theorem that allows them
to evaluate a definite integral using an antiderivative of the integrand, they often have
difficulty dealing with the concept of a function defined by an integral. We give some
exercises here that rely on the evaluation part of the FTC to help understand the
antiderivative part of the FTC.
Example 1
x
Define g( x) = ∫ sin( t)
dt. What is g′
( x) ?
0
Solution: Since −cos( t ) is an antiderivative for sin( t ), we have
Thus g′ ( x) = sin( x).
x
x
∫ 0
0
g( x) = sin( t) dt = − cos( t) = − cos( x)
+ 1.
Other examples in which the students can carry out the antidifferentiation may help them
x
see that in fact the equation g( x) = ∫ f ( t)
dt defines a function. It is also helpful to change
a
the lower limit of integration while considering the same integrand to see that varying the
lower limit changes the new function by a constant and so does not affect the derivative.
Now consider the case in which the students cannot antidifferentiate the integrand.
Note that the following discussion assumes that the integrand does in fact have an
antiderivative. The fact that every continuous function does have an antiderivative is
one form of the FTC that is proven in Section IV of these materials. What we are hoping
to accomplish here is to further students’ understanding of the FTC statements.
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AP® Calculus: 2006–2007 Workshop Materials