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Multivariable Calculus

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Difficult

Change of Variable According to Boundary

MVCALC-MUKMMY

Evaluate $I=\iint_R (x^2+y^2)dxdy$, where $R$ is the region located in the first quadrant with boundaries on curves:

$$ a^2 y=x^3, \quad b^2 y=x^3, \quad p^2 x=y^3, \quad q^2 x=y^3, \quad 0 < a < b, 0 < p < q $$

A

$\frac{2}{15} (b^{\frac{5}{2}}-a^\frac{5}{2})(q^{\frac{3}{2}}-p^{\frac{3}{2}})$

B

$\frac{2}{15} (b^{\frac{3}{2}}-a^{\frac{3}{2}}) (q^{\frac{5}{2}}-p^\frac{5}{2})$

C

$\frac{2}{15} (b^{\frac{5}{2}}-a^\frac{5}{2})(q^{\frac{3}{2}}-p^{\frac{3}{2}}) + \frac{2}{15} (b^{\frac{3}{2}}-a^{\frac{3}{2}}) (q^{\frac{5}{2}}-p^\frac{5}{2})$

D

$\frac{1}{15}(b^{\frac{3}{2}}-a^\frac{3}{2})(q^{\frac{3}{2}}-p^{\frac{3}{2}}) + \frac{1}{15}(b^{\frac{5}{2}}-a^{\frac{5}{2}}) (q^{\frac{5}{2}}-p^\frac{5}{2})$

E

$(b-a)^2+(q-p)^2$