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

∬(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