How would I integrate this:
∬R xy√(x^2 + y^2) R = [0,1]x[0,1]
I really have no idea.
∬R xy√(x^2 + y^2) R = [0,1]x[0,1]
I really have no idea.
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∫∫R xy√(x^2 + y^2) dA on [0,1] x [0,1]
= ∫(0 to 1) ∫(0 to 1) xy√(x^2 + y^2) dy dx
= ∫(0 to 1) x/3*(x^2 + y^2)^(3/2) eval. from y = 0 to y = 1 dx
= ∫(0 to 1) x/3 * (x^2 + 1)^(3/2) - (x^4)/3 dx
= 1/15 * (x^2 + 1)^(5/2) - (x^5)/15 eval. from 0 to 1
= 1/15 * (2)^(5/2) - 1/15 - 1/15 = 0.38 - 2/15 = 0.24379
= ∫(0 to 1) ∫(0 to 1) xy√(x^2 + y^2) dy dx
= ∫(0 to 1) x/3*(x^2 + y^2)^(3/2) eval. from y = 0 to y = 1 dx
= ∫(0 to 1) x/3 * (x^2 + 1)^(3/2) - (x^4)/3 dx
= 1/15 * (x^2 + 1)^(5/2) - (x^5)/15 eval. from 0 to 1
= 1/15 * (2)^(5/2) - 1/15 - 1/15 = 0.38 - 2/15 = 0.24379