Question

【高等数学】第6题题目见图,求几阶无穷小量

Answer 1

I =∫∫ [1-cos(x^2+y^2)]dxdy

=∫dθ∫ [1-cos(r^2)]rdr

= π∫ [1-cos(r^2)]dr^2

= π[r^2-sin(r^2)] = π[t^2-sin(t^2)] .

lim π[t^2-sin(t^2)]/t^k

= lim π[2t-2tcos(t^2)]/[kt^(k-1)]

= lim 2π[1-cos(t^2)]/[kt^(k-2)]

= lim π(t^2)^2/[kt^(k-2)] = C,则 k=6.