Science:Math Exam Resources/Courses/MATH101/April 2018/Question 11 (ii)/Statement

Since we all love calculus, lastly we will consider the heart-shaped region ${\textstyle {\mathcal {H}}}$ bounded by

$$\displaystyle x^{2}+\left({\frac {5y}{4}}-{\sqrt {|x|}}\right)^{2}=1\,.$$

The plot of ${\textstyle {\mathcal {H}}}$ is shown below (where the coordinate axes have been deliberately removed).

(ii) Assume that the density of the heart-shaped region is constant. Show that the ${\textstyle y}$-coordinate of the centroid of this region, labelled by ${\textstyle {\bar {y}}}$, can be written in the form ${\textstyle {\bar {y}}=b\int _{0}^{1}{\sqrt {x-x^{3}}}\,dx}$. Determine the constant ${\textstyle b}$ explicitly.