The curve defined byx=acos^{3}t,y=asin^{3}tfor 0 ≤t≤ 2π,a≠ 0, is known as an astroid.

The curve could be approximated by the arcs of four quadrants each of radiusa.

By considering the coordinates of particular points on the curve and on the approximating arcs of the quadrants for corresponding values oft, determine whether the total length of the astroid is greater or less than the sum of the arcs of the quadrants.

Find the total (exact) length of the asteroid by integration.

Hence confirm the approximation described above.

Establish the integral required to find the surface area formed when a parametric curve

x= f(t),y= g(t) is rotated about thex-axis.

Find the surface area formed when the curvex=acos^{3}t,y=asin^{3}t(0 ≤t≤ π/2) is rotated about thex-axis.

Compare the surface area obtained with the surface area of a hemisphere of radiusa.

From your work in part a), why might some people expect the surface area found above to be similar to the surface area of a hemisphere, and why are they actually rather different?