Let be the curve where .

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Hi, Maham,

We can find this by solving the integral:

`int_a^bsqrt(1+(dy/dx)^2)dx` from 1 to 4

So, first, we need to find the derivative of y. That is:

5*x^1.5 --> **7.5 x^0.5**

Then, we plug that into the integral. First, we can simplify under the square root sign:

1 + (7.5x^0.5)^2 = **1 + 56.25x**

So, we have:

`int_a^bsqrt(1 + 56.25x)dx` from 1 to 4

The integral of this is: **(2/168.75)(1+56.25x)^1.5**

Plugging in 4 for x, we get:

(2/168.75)(1+56.25*4)^1.5 = **40.27**

Then, plugging in 1 for x:

(2/168.75)(1+56.25*1)^1.5 = **5.13**

Subtracting these values, the length of the curve is approx.:

40.27 - 5.13 = **35.14** units.

Good luck, Maham. I hope this helps.

Till Then,

Steve

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