THE DEFINITE INTEGRAL AND THE FUNDAMENTAL THEOREM OF CALCULUS
155
20.26
When
and, when x = 5, u = 121 Then
20.27
20.28
Then
Then
Then
20.29
20.30
20.31
20.32
Find the average value of
By definition, the average value of a function f(x) on an interval [a, b] is
Hence, we must
compute
20.33
Compute the average value of f(x) = sec
2 x on [0,7T/41.
The average value is
20.34
State the Mean-Value Theorem for integrals.
If a function/is continuous on [a, b], it assumes its average value in [a, b]; that is,
for some c in [a, b].
20.35
Verify the Mean-Value Theorem for integrals, for the function f(x) = x + 2 on [1,2].
and
But, I = x + 2 when x = |.
20.36
Verify the Mean-Value Theorem for integrals, for the function f(x) = x
3
on [0,1].
But
when
and
on [0,1].
Let u = x
3 -4, x
3 = u + 4, du = 3jc
2 dx.
x = 2, u = 4,
Let w = ;c
2 -9, x
2 = u+9, du = 2xdx.
Let u=2x
2 + l, du=4xdx.
Let M = x + l, x = «-l, du = dx.
S"fMdx=f(c)(c)