Region in the –plane is bounded by the –axis

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

Problem

Region

S

in the

xy

–plane is bounded by the

x

–axis, the

y

–axis, and the line

y=2-\dfrac{x}{2}

. Two solids are constructed on the same base

S

. • Solid P has cross sections perpendicular to the

x

–axis that are semicircles whose diameters lie in the plane of

S

. • Solid Q has cross sections perpendicular to the

x

–axis that are equilateral triangles whose bases lie in the plane of

S

. Solid P is made of a material with constant density

\rho_{P}

(in grams per cubic centimeter), and Solid Q is made of a different material with constant density

\rho_{Q}

. (a) Without performing any integrations, determine which solid has the greater volume. Justify your conclusion by analyzing the relative growth rates of the cross-sectional areas along the

x

–axis. (b) Find an inequality involving

\rho_{P}

and

\rho_{Q}

that guarantees Solid P is heavier than Solid Q. Clearly state your reasoning. (c) Assume

\rho_{P}=1.2\text{ g/cm}^{3}

and

\rho_{Q}=0.9\text{ g/cm}^{3}

. Decide whether Solid P or Solid Q has the greater mass and support your conclusion without computing definite integrals.

Related theory

Browse the problem library