04/19/23 Perkins

AP Calculus AB

Day 10Section 4.4

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

Perkins

AP Calculus AB

Day 10Section 4.4

The Mean Value Theorem for Integrals

a b

4

2

5

1. Find the average value of on [1,4]. Then find where f(x) obtains this average value.

23 2y x x

2

2

2. 3xdt t dt

dx

Second Fundamental Theorem of Calculus

a

u

f t dtd

dx

3

33.xdt t dt

dx