Let be the region bounded on the right by the curve defined implicitly by x^ 2 +xy+y^…

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

Let

D

be the region bounded on the right by the curve defined implicitly by

x^{2}+xy+y^{2}=3y,

and on the left by the

y

-axis. (a) Solve the implicit equation for

x

in terms of

y

. Show all algebraic steps, and indicate which branch corresponds to the rightmost part of the curve. (b) Find all

y

-values at which the curve meets the

y

-axis. Justify your result. (c) Formulate a definite integral with respect to

y

whose value equals the area of the region

D

. Clearly identify the integrand and limits of integration. (d) Evaluate the integral obtained in part (c) to compute the exact area of

D