Question

二重积分xy^2dxdy,D是由抛物线y^2=2px和直线x=p/2围成的区域（p>0）

Answer 1

∫∫(D)xy²dxdy=∫(0,p/2)xdx∫(-√(2px),√(2px))y²dy

=∫(0,p/2)x{[y³/3]│(-√(2px),√(2px))}dx

=∫(0,p/2)x[2√(2px)³/3]dx

=[2^(5/2)*p^(3/2)/3]∫(0,p/2)x^(5/2)dx

=[2^(5/2)*p^(3/2)/3]*[(2/7)*(p/2)^(7/2)]

=p^5/21.