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Mathematics
Problem
A function
h
h
h
is defined by
h
(
x
)
=
{
3
x
,
0
≤
x
<
2
,
12
−
x
2
,
2
≤
x
≤
5.
h(x)=\begin{cases} 3x, & 0\le x<2,\\ 12-x^{2}, & 2\le x\le 5. \end{cases}
h
(
x
)
=
{
3
x
,
12
−
x
2
,
0
≤
x
<
2
,
2
≤
x
≤
5.
Find the average value of
h
h
h
on the interval
[
1
,
4
]
[1,4]
[
1
,
4
]
. (a) Evaluate
∫
1
4
h
(
x
)
d
x
\displaystyle \int_{1}^{4} h(x)\,dx
∫
1
4
h
(
x
)
d
x
. (b) Using your result from part (a), determine the average value of
h
h
h
over
[
1
,
4
]
[1,4]
[
1
,
4
]
. Express your answer in exact form.
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