find the length of the arc of the curve from point P to point Q. y=1/2x^2, P(-7,49/2), Q(7,49/2) please explain the "why" as you solve

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The graph is a parabola:

The parabola is symmetric, so we may instead find the length of the curve from (0,0) to (7, 49/2) and double it:

A formula for arc length is:

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

Thus we want to find:

`2 int _0 ^7 sqrt(1+(dy/dx)^2) dx`

`y=(1/2)x^2`

`dy/dx = x`

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

So we want to find:

`2 int_0^7 sqrt(1+x^2) dx`

We ignore the constant, and the definite integral for now:

`int sqrt(1+x^2)dx`

We can make the trig substitution:

`x="tan" theta` , `dx = "sec"^2 theta d theta`

The integral becomes:

`int sqrt(1+"tan"^2 theta) "sec"^2 theta d theta`

`=int "sec"^3 theta d theta`

This is an integration by parts problem:

`u= "sec" theta`            ...

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