A solid has as its base the region in the first quadrant bounded by the curve

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

Problem

y=\sqrt{x}

, the line

y=2

, and the

y

-axis. Each cross section of the solid taken perpendicular to the

y

-axis is an equilateral triangle whose base lies in the region and whose altitude is perpendicular to the base. The coordinate units are centimeters. (a) Express the length

s(y)

of the side of the triangular cross section as a function of

y

for

0\le y\le2

. Show all reasoning, including any algebraic manipulation used to obtain

s(y)

. State appropriate units. (b) Write, but do NOT evaluate, a definite integral whose value equals the volume

V

of the solid. Clearly indicate limits of integration and all functions used. (c) Evaluate the integral from part (b) to obtain an exact value for

V

. Include units in your final answer. (d) Briefly interpret the meaning of the integrand in part (b) in terms of geometry of the solid. Your answer should reference both the base region and the cross-sectional shape.

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