Find Length of Curve y=(2/3)(x^2+1)^(3/2)

[whatlifeforme]

Homework Statement

find the length of the following curve.

Homework Equations

y=(2/3)(x^2 +1)^(3/2) from x=3 to x=9.

The Attempt at a Solution

f'(x) = 2x^3 + 2x
f'(x)^2 = 4x^6 + 8x^4 + 4x^2

L = integral (3,9) sqrt(1+4x^6 + 8x^4 + 4x^2)

 

[tiny-tim]

hi whatlifeform!

y=(2/3)(x^2 +1)^(3/2) from x=3 to x=9.
…
f'(x) = 2x^3 + 2x

no, you've got muddled

try the chain rule again ​

 

[whatlifeforme]

dy/dx = (2x) (x^2 + 1) ^ (1/2)
(dy/dx)^2 = 4x^2 (x^2+1)

L = integral (3,9) sqrt(1+4x^4 + 4x^2)

 

[haruspex]

dy/dx = (2x) (x^2 + 1) ^ (1/2)
(dy/dx)^2 = 4x^2 (x^2+1)

L = integral (3,9) sqrt(1+4x^4 + 4x^2)

Yes. Can you see a simplification?

 

[tiny-tim]

ie, what is √(4x⁴ + 4x² + 1) ?

 

[whatlifeforme]

ie, what is √(4x⁴ + 4x² + 1) ?

take the integral of this?

 

[dx]

Write the expression under the square-root as the square of something.