Answer
$\dfrac{243}{8}$
Work Step by Step
$I=\int_C (x^2y^3-\sqrt x) dy=\int_C ((y^2)^2 y^3-y) dy$
$=\int_{1}^{2}y^7-y dy$
$=[\dfrac{y^8}{8}-\dfrac{y^2}{2}]_1^2$
$=[\dfrac{2^8}{8}-\dfrac{2^2}{2}]-[\dfrac{1}{8}-\dfrac{1}{2}]$
$=[\dfrac{256}{8}-\dfrac{16}{8}]-[\dfrac{1}{8}-\dfrac{4}{8}]$
$= \dfrac{243}{8}$
