Answer
$\dfrac{9}{14}$
Work Step by Step
Consider $f(x)= x^3+\sqrt y=1$
Re-write as: $y=(1-x^3)^2=x^6-2x^3+1$
Use formula as follows: $\int x^{n} dx=\dfrac{x^{n+1}}{n+1}+C$
This implies that
$[\dfrac{x^{6+1}}{6+1}-\dfrac{2x^4}{4}+x]_0^1=(\dfrac{1}{7}-0)-2(\dfrac{1}{4}-0)+(1-0)$
Thus, $f(x)=\int_0^1 x^6-2x^3+1 dx=\dfrac{9}{14}$
