Mathematics

Problem

A circle is defined implicitly by the equation

x^{2}+y^{2}=25.

Let

R

be the portion of the circle that lies to the right of the

y

-axis. (a) Solve the equation

x^{2}+y^{2}=25

for

x

in terms of

y

. Express your answer in simplest radical form. (b) Explain why the expression obtained in part (a) must be restricted to a single branch when

x

is to be treated as a function of

y

for computing the area of

R

with a single definite integral. (c) Identify which branch from part (a) correctly represents the curve bounding region

R

, and state the corresponding interval of

y

-values over which this branch is valid.