Alice starts at the origin and walks along the graph of $y=\frac{x^{2}}{2}$ at a velocity of 6 units/second.Which of the following integrals show how far Alice has traveled when she reaches the point where x = a?
A) $\int_{0}^{a \sqrt{6}} \sqrt{1+x^{2}} d x$
B) $\int_{0}^{a} \sqrt{1+x^{2}} d x$
C) $\int_{0}^{6 a} \sqrt{1+x^{2}} d x$
D) $6 \int_{0}^{a} \sqrt{1+x^{2}} d x$

Question

Quizplus · Accepted Answer

The distance traveled along a curve $y=f(x)$ from $x=0$ to $x=a$ can be found using the integral $\int_{0}^{a} \sqrt{1+(f'(x))^2} dx$. For $y=\frac{x^2}{2}$, $f'(x)=x$, so the integral becomes $\int_{0}^{a} \sqrt{1+x^2} dx$, matching option B.

Alice Starts at the Origin and Walks Along the Graph y=x22y=\frac{x^{2}}{2}y=2x2​

Alice Starts at the Origin and Walks Along the Graph $y=\frac{x^{2}}{2}$