the length of the curve y = x^4 from x = 1 to x = 5 is given by (a) ∫₁⁵√(1 + 4x³) dx (b) ∫₁⁵√(1 + x⁴) dx (c)…

Explanation:

Step1: Recall arc - length formula

The arc - length formula for a function $y = f(x)$ from $x=a$ to $x = b$ is $L=\int_{a}^{b}\sqrt{1+(y')^{2}}dx$.

Step2: Differentiate $y = x^{4}$

If $y=x^{4}$, then by the power rule $(x^{n})'=nx^{n - 1}$, we have $y' = 4x^{3}$.

Step3: Calculate $(y')^{2}$

$(y')^{2}=(4x^{3})^{2}=16x^{6}$.

Step4: Substitute into arc - length formula

Substitute $(y')^{2}=16x^{6}$ into the arc - length formula $L=\int_{a}^{b}\sqrt{1+(y')^{2}}dx$. Here $a = 1$, $b = 5$, so $L=\int_{1}^{5}\sqrt{1 + 16x^{6}}dx$.

Answer:

D. $\int_{1}^{5}\sqrt{1+16x^{6}}dx$