Verified answer

1) dy/dx = √(x^2 - 1) by the Fundamental Theorem of Calculus.

So, √(1 + (dy/dx)^2)

= √(1 + (x^2 - 1))

= √(x^2)

= x, since x is in [4, 7] (and thus positive).

Hence, the arc length equals

∫(x = 4 to 7) x dx

= (1/2)x^2 {for x = 4 to 7}

= 33/2.

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2) √(1 + (dy/dx)^2)

= √(1 + (3x^(1/2))^2)

= √(1 + 9x).

Hence, the arc length equals

∫(x = 0 to 5/4) (1 + 9x)^(1/2) dx

= (1/9)(2/3)(1 + 9x)^(3/2) {for x = 0 to 5/4}

= (2/27) [(49/4)^(3/2) - 1]

= 335/108.

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I hope this helps!

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