How do you find the average distance from the origin of a point on the parabola y=x2,0≤x≤4 with respect to x?
Answers
Answer:
17
√
17
−
1
12
≈
5.84
Explanation:
Let
P
be a point on the parabola. The coordinates with respect to
x
will be
(
x
,
x
2
)
.
We can make a function out of the distance to the origin using the distance formula:
d
=
√
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
Applying this to our point
P
=
(
x
,
x
2
)
and the origin
(
0
,
0
)
, we get:
d
(
x
)
=
√
(
x
−
0
)
2
+
(
x
2
−
0
)
2
d
(
x
)
=
√
x
2
+
x
4
To work out the average value of the function, we can use the average value formula. For a function
f
(
x
)
on the interval
[
a
,
b
]
, the average value will be:
∫
b
a
f
(
x
)
d
x
b
−
a
In our case, we get:
∫
4
0
d
(
x
)
d
x
4
−
0
=
1
4
∫
4
0
√
x
2
+
x
4
d
x
I will first work out the antiderivative:
∫
√
x
2
+
x
4
d
x
=
=
∫
√
x
2
(
1
+
x
2
)
d
x
=
=
∫
x
√
x
2
+
1
d
x
=
Now we can introduce a u-substitution with
u
=
x
2
+
1
. The derivative of
u
is then
2
x
, so we divide through by that:
=
∫
x
√
u
2
x
d
u
=
1
2
∫
√
u
d
u
=
1
2
⋅
2
3
u
3
2
+
C
=
2
6
(
1
+
x
2
)
3
2
+
C
Now we can evaluate the original expression:
1
4
∫
4
0
√
x
4
+
x
2
d
x
=
1
4
[
1
3
(
1
+
x
2
)
3
2
]
4
0
=
=
1
4
(
1
3
(
1
+
16
)
3
2
−
1
3
(
1
+
0
)
3
2
)
=
=
1
4
(
1
3
⋅
17
3
2
−
1
3
⋅
1
3
2
)
=
1
4
(
17
√
17
3
−
1
3
)
=
=
1
4
⋅
17
√
17
−
1
3
=
17
√
17
−
1
12
So, the average distance from a point to the origin on the parabola on the interval
[
0
,
4
]
is:
17
√
17
−
1
12
≈
5.84