Find positive numbers a and b so that the change of variables s=ax,t=by transforms the integral ∫∫rdxdy into ∫∫t∣∣∣∂(x,y)∂(s,t)∣∣∣dsdt for the region r, the rectangle 0≤x≤15, 0≤y≤60 and the region t, the square 0≤s,t≤1. a= b= what is ∣∣∂(x,y)∂(s,t)∣∣ in this case? ∣∣∂(x,y)∂(s,t)∣∣=
Accepted Solution
A:
s = ax It is required to find the value of a which make: 0 ≤ s ≤ 1 when 0 ≤ x ≤ 15 ∴ s = 0 when x = 0 ⇒⇒⇒ a = 0 (unacceptable) or s = 1 when x = 15 ⇒⇒⇒ a = s/x = 1/15
Similarly, t = by It is required to find the value of b which make: 0 ≤ t ≤ 1 when 0 ≤ y ≤ 60 ∴ t = 0 when y = 0 ⇒⇒⇒ b = 0 (unacceptable) or t = 1 when y = 60 ⇒⇒⇒ b = t/y = 1/60