Region consists of two adjoining subregions situated above the –axis

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

Problem

Region

R

consists of two adjoining subregions situated above the

x

–axis. • For

-2 \le x \le 0

, the upper boundary of

R

is the line

y = x + 2

. • For

0 \le x \le 2

, the upper boundary of

R

is the semicircle

y = \sqrt{4 - x^2}

. When the entire region

R

is revolved about the horizontal line

y = 3

, a solid

S

is created. (a) Verify analytically that the two boundary curves meet at the point

(0,2)

. (b) Write separate definite integrals that together represent the volume of

S

. Do not evaluate at this stage. (c) Evaluate each integral from part (b) and determine the exact total volume of

S

. (d) State appropriate units for the volume and briefly interpret the physical meaning of your result.