A river cross-section is modeled in the -plane

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

Problem

A river cross-section is modeled in the

xy

-plane. From the riverbed at

y=-1

up to the water surface at

y=6

, the submerged left boundary is given by

x=(y-1)^{3}-4

. The right boundary is given by

x=5-\dfrac{y^{2}}{2}

for

-1\le y\le4

. Above

y=4

the riverbank is a vertical retaining wall at

x=1

, so for

4<y\le6

the right boundary is the line

x=1

. The region between the boundaries, the riverbed, and the water surface is denoted

U

. (a) Verify that the curves

x=(y-1)^{3}-4

and

x=5-\dfrac{y^{2}}{2}

intersect exactly once on the interval

-1\le y\le4

, and compute the

y

-coordinate of this point of intersection. (b) Write an expression for the total area of

U

as the sum of one or more definite integrals with respect to

y

. Clearly indicate the limits of integration for each integral. (c) Evaluate each integral from part (b) to determine the exact cross-sectional area of the river. (d) Interpret the meaning of your result from part (c) in the context of the river cross-section, including appropriate units.

Related theory

Browse the problem library