Question

what is the value of double integration R 3xy² dxdy where R is the region bounded by curve y²=x and y=x³

Answer 1

\(\displaystyle\iint\limits_R 3\mathrm xy^2d\mathrm x \,dy\)

R is region bounded by curve y² = x & y = x³.

\(\displaystyle\int\limits_0^1\left(\int^{\sqrt{\mathrm x}}_{\mathrm x^3}3\mathrm xy^2\,dy\right)d\mathrm x\) 

\(=\displaystyle \int\limits_0^13\mathrm x\left(\frac{y^3}{3}\right)^{\sqrt{\mathrm x}}_{\mathrm x^3}d\mathrm x\)

\(=\displaystyle\int\limits_0^1 \mathrm x\left(\mathrm x^{\frac{3}{2}}-\mathrm x^9\right)d\mathrm x\)

\(=\displaystyle\int\limits_0^1 (\mathrm x^{\frac{5}{2}}-\mathrm x^{10})d\mathrm x\)

\(=\frac{2}{7}(\mathrm x^{\frac{7}{2}})^1_0-\frac{1}{11}(\mathrm x^{11})^1_0\)

\(=\frac{2}{7}-\frac{1}{11}=\frac{22-7}{77}=\frac{15}{77}\)