Question

Oblicz całkę: ∬(x²-y²)dxdy
zbiór D ={(x,y); x²+y²<1 ∧ -x≤y≤0}

Answer 1

∬(x² - y²)dxdy

D = { (x,y); x² + y² < 1 ∧ - x ≤ y ≤ 0 }

x = rcost

y = rsint

0 < r < 1

0 < t < ½π

∬(x² - y²)dxdy = ∫∫[(rcost)² - (rsint)²]drdt =

= ∫∫[r²(cos²t - sin²t)]drdt =

= ∫∫[r²(1 - 2sin²t)]drdt =

= ∫r²dr * ∫(1 - 2sin²t)dt =

= 1/3*½π =

= 1/6π


∫r²dr = 1/3r³ = 1/3*(1 - 0) = 1/3


∫(1 - 2sin²t)dt = t - 2∫sin²tdt =

= t - 2( - sint*cost - cost) =

= t + sin2t + cost =

= (½π + sinπ + cosπ) - (0 + sin0 + cos0) =

= ½π - 1 + 1 =

= ½π