5.1. CONSTRUCTING ACCURATE GRAPHS OF ANTIDERIVATIVES
271
(c) original function: p(x) =
x 2 ,
if 0 < x ≤ 1
−(x − 2) 2 , if 1 < x < 2
0
otherwise
;
initial condition: P(0) = 1;
interval for sketch: [−1, 3]
⊳
Functions defined by integrals
In Equation (5.1), we found an important rule that enables us to compute the value of the
antiderivative F at a point b, provided that we know F(a) and can evaluate the definite
integral from a to b of f . Again, that rule is
F(b) = F(a) +
b
a
f (x) dx.
In several examples, we have used this formula to compute several different values of F(b)
and then plotted the points (b, F(b)) to assist us in generating an accurate graph of F.
That suggests that we may want to think of b, the upper limit of integration, as a variable
itself. To that end, we introduce the idea of an integral function, a function whose formula
involves a definite integral.
Given a continuous function f , we define the corresponding integral function A
according to the rule
A(x) =
x
a
f (t) dt.
(5.2)
Note particularly that because we are using the variable x as the independent variable
in the function A, and x determines the other endpoint of the interval over which we
integrate (starting from a), we need to use a variable other than x as the variable of
integration. A standard choice is t, but any variable other than x is acceptable.
One way to think of the function A is as the “net-signed area from a up to x” function,
where we consider the region bounded by y = f (t) on the relevant interval. For example,
in Figure 5.4, we see a given function f pictured at left, and its corresponding area function
(choosing a = 0), A(x) =
x
0
f (t) dt shown at right.
Note particularly that the function A measures the net-signed area from t = 0 to t = x
bounded by the curve y = f (t); this value is then reported as the corresponding height on
the graph of y = A(x). It is even more natural to think of this relationship between f and
A dynamically. At http://gvsu.edu/s/cz, we find a java applet 1 that brings the static
picture in Figure 5.4 to life. There, the user can move the red point on the function f and
see how the corresponding height changes at the light blue point on the graph of A.
1 David Austin, Grand Valley State University
271
(c) original function: p(x) =
x 2 ,
if 0 < x ≤ 1
−(x − 2) 2 , if 1 < x < 2
0
otherwise
;
initial condition: P(0) = 1;
interval for sketch: [−1, 3]
⊳
Functions defined by integrals
In Equation (5.1), we found an important rule that enables us to compute the value of the
antiderivative F at a point b, provided that we know F(a) and can evaluate the definite
integral from a to b of f . Again, that rule is
F(b) = F(a) +
b
a
f (x) dx.
In several examples, we have used this formula to compute several different values of F(b)
and then plotted the points (b, F(b)) to assist us in generating an accurate graph of F.
That suggests that we may want to think of b, the upper limit of integration, as a variable
itself. To that end, we introduce the idea of an integral function, a function whose formula
involves a definite integral.
Given a continuous function f , we define the corresponding integral function A
according to the rule
A(x) =
x
a
f (t) dt.
(5.2)
Note particularly that because we are using the variable x as the independent variable
in the function A, and x determines the other endpoint of the interval over which we
integrate (starting from a), we need to use a variable other than x as the variable of
integration. A standard choice is t, but any variable other than x is acceptable.
One way to think of the function A is as the “net-signed area from a up to x” function,
where we consider the region bounded by y = f (t) on the relevant interval. For example,
in Figure 5.4, we see a given function f pictured at left, and its corresponding area function
(choosing a = 0), A(x) =
x
0
f (t) dt shown at right.
Note particularly that the function A measures the net-signed area from t = 0 to t = x
bounded by the curve y = f (t); this value is then reported as the corresponding height on
the graph of y = A(x). It is even more natural to think of this relationship between f and
A dynamically. At http://gvsu.edu/s/cz, we find a java applet 1 that brings the static
picture in Figure 5.4 to life. There, the user can move the red point on the function f and
see how the corresponding height changes at the light blue point on the graph of A.
1 David Austin, Grand Valley State University