[!!]. This exercise is used in Section e12.6.

(a) Prove that the improper integral Jooo e-x2 dx converges to a finite real number.

(b) Prove that if 1 is the value of the integral in part (a), then 17r/21N 12 = lim e-r2 r dr dB.

N-->oo 0 0

(c) Show that roo e-x2 dx = y'7f.

io 2

(d) Let Qk represent the n-dimensional cube [-k, k] x ... x [-k, k]. Find lim r e-llzl12 dz.