Find the exact length of the curve y=2/3(x^2-1)^(3/2),  1 ≤ x ≤ 3  i dont know ho to start off can someone help me

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Notice that the curve whose length you should find is defined by a rectangular equation, hence, you need to use the next formula such that:

`int_a^b sqrt(1 + ((dy)/(dx))^2)dx`

You need to differentiate the given curve using the chain rule such that:

`((dy)/(dx)) = (2/3)*(3/2)*(x^2 - 1)^(3/2 - 1)*(x^2 - 1)'`

`((dy)/(dx)) = 2x*sqrt(x^2 - 1)`

You need to raise to square such that:

`((dy)/(dx))^2 = 4x^2*(x^2 - 1)`

You need to add 1 such that:

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

`((dy)/(dx))^2 + 1 = (2x^2 - 1)^2`

Taking the square root both sides yields:

`sqrt(1 + ((dy)/(dx))^2) = sqrt((2x^2 - 1)^2)`

`sqrt(1 + ((dy)/(dx))^2) = |2x^2 - 1|`

You may evaluate the length of the curve such that:

`int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = int_1^3 (2x^2 - 1)dx`

`int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = int_1^3 (2x^2)dx - int_1^3 dx`

`int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = (2x^3/3 - x)|_1^3`

`int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 54/3 - 3 - 2/3 + 1`

`int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 52/3 - 2 `

`int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 46/3`

Hence, evaluating the length of the given curve yields `ds = int_1^3 sqrt(1 + ((dy)/(dx))^2)dx = 46/3` .

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