For constants and

MATHEMATICS · Single correct · AP · Calculus BC · Applications of Integration · Applications of Integration

Problem

For constants

a>0

and

k>0

, the curve

y=kx^{2}

is considered on

[-a,a]

. Which of the following gives the exact length of this curve? (a)

\displaystyle\frac{a}{2}\sqrt{1+4k^{2}a^{2}}+\frac{1}{4k}\operatorname{arcsinh}(2ka)

(b)

\displaystyle a\sqrt{1+4k^{2}a^{2}}+\frac{1}{2k}\operatorname{arcsinh}(2ka)

(c)

\displaystyle\frac{a}{k}\sqrt{1+4k^{2}a^{2}}+\frac{1}{2k}\operatorname{arcsinh}(2ka)

(d)

\displaystyle a\sqrt{1+k^{2}a^{2}}+\frac{1}{k}\operatorname{arcsinh}(ka)