The length of the curve \(y=x^4\) from \(x=1\) to \(x=5\) is given by

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The arc length for a curve is given by the equation:

$$ L = \int\limits_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}   dx $$ $$ L = \int\limits_1^5 \sqrt{1+(4x^3)^2}   dx $$ $$ =\boxed{\int\limits_1^5 \sqrt{1+16x^6}   dx} $$