The curves with equations $x^n+y^n=1,\;n=4,6,8,\ldots,$ are called fat circles . (You should graph some of these curves to see why.)

(a) Which of the following integrals represents the length $L_{2k}$ of the fat circle with $n = 2k$ ?

A. $\displaystyle 4 \int_{0}^{\,1} \sqrt{1+\left[-x^{2k}\left(1-x^{2k}\right)^{1/(2k)}\right]^2}\,dx$
B. $\displaystyle 2 \int_{0}^{\,1} \sqrt{1+\left[-x^{2k-1}\left(1-x^{2k}\right)^{1/(2k)-1}\right]^2}\,dx$
C. $\displaystyle 4 \int_{0}^{\,1} \sqrt{1+\left[-x^{2k-1}\left(1-x^{2k}\right)^{1/(2k)-1}\right]^2}\,dx$
D. $\displaystyle 2 \int_{0}^{\,1} \sqrt{1+\left[-x^{2k}\left(1-x^{2k}\right)^{1/(2k)}\right]^2}\,dx$
E. $\displaystyle 2 \int_{0}^{\,1} \sqrt{1+\left[-x^{2k-1}\left(1-x^{2k}\right)^{1/(2k)}\right]^2}\,dx$
F. $\displaystyle 4 \int_{0}^{\,1} \sqrt{1+\left[-x^{2k-1}\left(1-x^{2k}\right)^{1/(2k)}\right]^2}\,dx$

(b) Without attempting to evaluate the correct integral in (a), state the value of $\displaystyle \lim_{k \to \infty} L_{2k}.$

$\displaystyle \lim_{k \to \infty} L_{2k} =$