Question

【高等数学】第6题题目见图，求几阶无穷小量 求具体解题过程！！答案6为什么？

Answer 1

I =∫∫ <x^2+y^2≤t^2> [1-cos(x^2+y^2)]dxdy
=∫<0,2π>dθ∫ <0, t> [1-cos(r^2)]rdr
= π∫ <0, t> [1-cos(r^2)]dr^2
= π[r^2-sin(r^2)] <0, t> = π[t^2-sin(t^2)] .
lim<t→0> π[t^2-sin(t^2)]/t^k
= lim<t→0> π[2t-2tcos(t^2)]/[kt^(k-1)]
= lim<t→0> 2π[1-cos(t^2)]/[kt^(k-2)]
= lim<t→0> π(t^2)^2/[kt^(k-2)] = C，则 k=6。