Region is bounded above by the line

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

Problem

Region

U

is bounded above by the line

y=2x

, below by the

x

–axis, and on the right by the vertical line

x=5

. Two solids share

U

as their base. • Solid M has cross sections taken perpendicular to the

x

–axis that are squares. • Solid N has cross sections taken perpendicular to the

x

–axis that are rectangles whose height at each

x

h(x)=a-x

, where

a

is a positive real constant (with

a>5

so that

h(x)

is positive on

[0,5]

). Answer each part in exact form. (a) Write an explicit definite integral for the volume

V_{M}

of Solid M in terms of

x

. Do not evaluate yet. (b) Write an explicit definite integral for the volume

V_{N}

of Solid N in terms of the parameter

a

. (c) Evaluate both integrals to obtain

V_{M}

and

V_{N}

as functions of

a

. (d) Determine the value of

a

for which the two volumes are equal, i.e., solve

V_{N}=V_{M}

. Show all work leading to your result.