The curve defined implicitly by encloses a symmetric figure known as an astroid

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

The curve defined implicitly by

x^{\,2/3}+y^{\,2/3}=1

encloses a symmetric figure known as an astroid. Let R denote the portion of the astroid in the right half–plane (

x\ge 0

). (a) Express

x

explicitly as a function of

y

for the right half–plane portion of the astroid, and state the allowable interval for

y

. (b) Derive a definite integral with respect to

y

whose value equals the area of R. (c) Evaluate the integral obtained in part (b) to show that

\displaystyle\text{Area}(R)=\frac{3\pi}{16}

. (d) Hence determine the total area enclosed by the astroid.