A circle with center and radius is described by (x-2)^ 2 +y^ 2 =4

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

Problem

A circle with center

(2,0)

and radius

2

is described by

(x-2)^{2}+y^{2}=4.

A parabola opens to the right and is given by

x=y^{2}.

Let

E

be the finite region enclosed between the right half of the circle and the parabola. (a) Express

x

explicitly as a function of

y

for the right half of the circle. State any domain restrictions on

y

that arise from this choice. (b) Determine the exact

y

-coordinates of the points where the circle and the parabola intersect. (c) Construct a single definite integral with respect to

y

whose value equals the area of the region

E

. Clearly indicate which function represents the right boundary and which represents the left boundary of the region. (d) Evaluate the integral from part (c) to obtain the exact area of

E

. Show all work leading to your result. (e) Briefly interpret how the sign of the integrand in part (c) confirms that the integrand represents a horizontal thickness throughout the interval of integration.

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