Section 6.6
1
Average value of a continuous function
We know that a discrete function with inputs
x1, x2 ,…, xn has outputs f (x1 ), f (x2 ),…, f (xn ). The
average value of the function is therefore the
average of these n numbers:
€
€ the discrete function f (x)
average value of
f (x1 ) + f (x 2 ) +L + f (x n )
=
n
€
But if the function f (x) is continuous, this method
won’t work: there are infinitely many values of the
€
function, so how can we add them up and divide by
however many there are?!
€
We can estimate the average value of the
continuous function f (x) over the interval a ≤ x ≤ b
by means of an integral: if we dissect the interval
a ≤ x ≤ b into n subintervals at the points
€
€
€
(a =)x0, x1, x2,K, xn (= b)
so that each subinterval has equal width
€
Δx =
€
b−a
,
n
Section 6.6
2
then the right rectangle approximation method
says that
∫ab f (x)dx ≈ f (x1 ) ⋅ Δx + f (x2 ) ⋅ Δx + L+ f (xn ) ⋅ Δx
= f (x1 ) + f (x2 ) +L + f (xn ) ⋅ Δx
(
)
f (x ) + f (x ) +L + f (x )
=
⋅ ( Δx ⋅ n)
n
1
=
2
n
f (x1 ) + f (x2 ) +L + f (xn )
⋅(b − a)
n
Therefore,
€
f (x1 ) + f (x 2 ) +L + f (x n ) ∫ab f (x)dx
≈
.
n
b−a
As we use more and more rectangles, the integral
approximation by right rectangles improves in
€
accuracy.
But also, using more rectangles
corresponds to averaging more and more values of
the function on the interval a ≤ x ≤ b, so that, in the
limit as n → ∞, this discrete average approximates
better and better the average value of the
continuous function over the interval a ≤ x ≤ b:
€
€
€
Section 6.6
3
average value of the continuous function f (x)
∫ab f (x)dx
=
.
b−a
€
If we let y represent this average value, we can
write
€
y ⋅ (b − a) = ∫ ab f (x)dx
€
an equation which has a useful interpretation: on
the right side, the integral measures the area
€ curve y = f (x) over the interval a ≤ x ≤ b,
under the
and the quantity on the left can be read as the area
of a rectangle with base on the interval a ≤ x ≤ b
and height y . In other words, y is€the height of the
€ on the interval a ≤ x ≤ b whose area equals
rectangle
the area under the curve y = f (x).
€
€
€
€
€
Section 6.6
€
4
We can use this idea of averaging a continuous
function to return to an idea we investigated a
while ago: the average value of the rate of change
of the function f (x) over the interval a ≤ x ≤ b we
f (b) − f (a)
know how to calculate as
. But if we do
b−a
not have
€ values of f (a) and
€ a way to compute the
f (b) directly, but instead have only information
about the rate function f ′(x), we can still compute
€
this average value by using the Fundamental
€
b
′
Theorem. Since ∫a f (x)dx = f (b) − f (a), we obtain:
€
∫ab f ′(x)dx
average rate of change of f (x) =
b−a
€
€
€