Show that if ƒ is continuous on [a, b], a Z b , and if
EXAMPLE 2 Show that if ƒ is continuous on [a, b], a Z b , and if
ƒsxd dx = 0 , L a
then ƒsxd = 0 at least once in [a, b].
Solution
The average value of ƒ on [a, b] is
avsƒd = 1 ƒsxd dx = 1 # 0=0.
b-a L a b-a
By the Mean Value Theorem, ƒ assumes this value at some point c H [a, b] .
Chapter 5: Integration
Fundamental Theorem, Part 1
If ƒ(t) is an integrable function over a finite interval I, then the integral from any fixed number a H I to another number x H I defines a new function F whose value at x is
Fsxd =
ƒstd dt . L (1)
For example, if ƒ is nonnegative and x lies to the right of a, then F(x) is the area under the graph from a to x (Figure 5.19). The variable x is the upper limit of integration of an inte-
y area
gral, but F is just like any other real-valued function of a real variable. For each value of y the input x, there is a well-defined numerical output, in this case the definite integral of ƒ
from a to x.
Equation (1) gives a way to define new functions, but its importance now is the con- nection it makes between integrals and derivatives. If ƒ is any continuous function, then the Fundamental Theorem asserts that F is a differentiable function of x whose derivative
b is ƒ itself. At every value of x,
FIGURE 5.19 The function F(x) defined
Fsxd =
ƒstd dt = ƒsxd .
by Equation (1) gives the area under the
dx
dx L a
graph of ƒ from a to x when ƒ is To gain some insight into why this result holds, we look at the geometry behind it. nonnegative and x 7 a .
If on ƒÚ0 [a, b], then the computation of F¿sxd from the definition of the derivative means taking the limit as h:0 of the difference quotient
Fsx + hd - Fsxd .
For h7 0, the numerator is obtained by subtracting two areas, so it is the area under the y graph of ƒ from x to (Figure 5.20). x+h If h is small, this area is approximately equal to the area of the rectangle of height ƒ(x) and width h, which can be seen from Figure 5.20. That is,
f (x)
Fsx + hd - Fsxd L h ƒsxd .
0 t a xx b
Dividing both sides of this approximation by h and letting h : 0, it is reasonable to expect that
FIGURE 5.20 In Equation (1), F(x) is the area to the left of x. Also, Fsx + hd is
Fsx + hd - Fsxd
lim
the area to the left of . The
x+h
F¿sxd =
= ƒsxd.
h :0
difference quotient [Fsx + hd - Fsxd] >h is then approximately equal to ƒ (x), the
This result is true even if the function ƒ is not positive, and it forms the first part of the height of the rectangle shown here.
Fundamental Theorem of Calculus.
THEOREM 4
The Fundamental Theorem of Calculus Part 1
If ƒ is continuous on [a, b] then x Fsxd = 1
a ƒstd dt is continuous on [a, b] and
differentiable on (a, b) and its derivative is ƒsxd ;
F¿sxd = d ƒstd dt = ƒsxd.
dx
L (2)
5.4 The Fundamental Theorem of Calculus
Before proving Theorem 4, we look at several examples to gain a better understanding of what it says.