Download - AP Calculus AB
The Mean Value Theorem for Integrals
b
a
b
f x
a
dx
Gives the average value of f(x) on [a,b].
a b
4
2
5
sum of all y-values
# avg
of y-value
y-values
1Average y-value
b
a
f xa
dxb
1. Find the average value of on [1,4]. Then find where f(x) obtains this average value.
Avg val
23 2y x x
4
2
1
13 2
4 1x x dx
43 2
1
1
3x x
164 16 0
3 1
483
16
This value is obtained when… 23 2 16x x 23 2 16 0x x
3 8 2 0x x 83 and 2x x
83...but only is in 1,4 .x
2
2
2. 3xdt t dt
dx
3 2 3 23 31 13 2 3 22 2
dx x
dx
3 2313 2 2
xdt t
dx
3 23 23 31 13 2 3 22 2
dx x
dx
2 3 0x x 2 3x x
23 1x x
2 3x x
2
2
3xdt t dt
dx
Using the 2nd Fundamental Theorem of Calculus:
Second Fundamental Theorem of Calculus
In order to use this shortcut we must have…
a
u
f t dtd
dx
fdu
udx
a derivative…
of an integral…whose lower bound is constant…and whose upper bound is a function of x.
3
33.xdt t dt
dx
Using the 2nd Fundamental Theorem of Calculus:
We don’t know how to integrate this function!
3 3 233x x x
2 9 33x x x