Suppose T=T(x,y), where T denotes temperature, and x and y denote distances in a Cartesian coordinate system. The quantity ψ(x,y)= ∂y 2
∂ 2
T
− ∂x
∂T
+ ∂y
∂T
is particularly important to an engineering application. Suppose we changed the coordinate system to (u,v) by u=x+y,v=x−y. Re-express ψ(x,y) in terms of derivatives of T with respect to u and v 2. Find ∬ D
dA, where D is the domain bounded by y=(x+1)/2, y=(x+4)/2,y=2−x and y=5−x. 3. Find ∬ D
x 2
+xydA, where D is the domain bounded by y=(x+1)/2, y=(x+4)/2,y=2−x and y=5−x. 4. Find I=∬ D
x 2
ydxdy where D={(x,y)∣1≤x 2
+y 2
≤4,y≥0} by using polar coordinates. 5. Sketch the cardioid curve r=a(1+sinθ ) (where a>0 ) and use double integration in polar coordinates to find the area inside it.
