Math, asked by Pandaaastha4040, 1 year ago

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

Answered by harshdeep1273
0

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

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