A region R is enclosed by the circle and the vertical line

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

Problem

A region R is enclosed by the circle

x^{2}+y^{2}=4

and the vertical line

x=1

. The region is to be described and analyzed using integrals with respect to

y

. (a) Solve

x^{2}+y^{2}=4

for

x

in terms of

y

and state the domain of

y

for each resulting branch. (b) For each admissible value of

y

, identify which branch of the circle forms the boundary of R that is farther from the

y

–axis. Justify your answer without reference to a graph. (c) Determine all

y

–values at which the circle intersects the line

x=1

. (d) Explain whether the area of R, calculated as a single integral in

y

, must be split into more than one integral. Provide a complete justification based on your work in parts (a)–(c).