Discussion
The length of the straight line \( y = 3x \) from \( x = 0 \) to \( x = 4 \) by the arc-length integral is
Answer: A
Here \( y' = 3 \), a constant, so the integrand is constant too:
\[ L \;=\; \int_{0}^{4}\sqrt{1 + 3^{2}}\,dx \;=\; \sqrt{10}\,\left[x\right]_{0}^{4} \;=\; 4\sqrt{10} \;\approx\; 12.65. \]
Pythagoras on the horizontal run of four and the vertical rise of twelve gives the same \( \sqrt{16 + 144} = \sqrt{160} \), confirming it
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