Let be the upper half of the circle of radius centered at the origin

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

Problem

Let

C

be the upper half of the circle of radius

3

centered at the origin, given by

y=\sqrt{9-x^{2}}

. Let

k(x)=\tfrac13 x^{2}

for

-3\le x\le 3

. The graphs of

y=\sqrt{9-x^{2}}

and

y=k(x)

enclose a lens-shaped region that is composed of four congruent parts. (a) Find the exact

x

-coordinates of all intersection points of the two graphs. (b) Express the total area of the lens-shaped region as a sum of definite integrals over appropriate intervals. Do not evaluate yet, but clearly indicate how many integrals are needed and why. (c) Evaluate the integral expression from part (b) to obtain the exact area of the lens-shaped region. (d) Suppose the region is cut from a uniform sheet of metal having surface density

0.5\ \text{kg}\,\text{m}^{-2}

. Determine the mass of the metal region, stating appropriate units.

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