An irrigation channel has vertical side walls at and a curved earthen bank on the…

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

Problem

An irrigation channel has vertical side walls at

x=0

and a curved earthen bank on the opposite side described by

x=6-y^{2}

for

-2\le y\le 2

, where

y

is measured in meters relative to the channel’s centerline and

x

is the horizontal distance from the left wall. Answer the following questions about the wetted cross–section. (a) Express the cross–sectional area

A

(in square meters) of water in the channel as a definite integral with respect to

y

. Do not evaluate the integral in this part. (b) Evaluate the integral obtained in part (a) to obtain the exact numerical value of

A

. (c) Suppose the water level rises so that the curved bank is now described by the same equation

x=6-y^{2}

, but for

-a\le y\le a

with

a>0

. Derive a simplified algebraic formula

A(a)

for the cross–sectional area in terms of

a

, and determine

\displaystyle\frac{dA}{da}

. Interpret the sign of

\frac{dA}{da}

in the physical context of the channel.