Mathematics

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.

Intuitive explanation

This problem asks you to find the area of a region bounded by curves that change shape as you move vertically—a classic setup for integration with respect to $y$ . The key insight is to recognize that the river's width (the horizontal distance between left and right boundaries) varies with height, so you'll slice the region horizontally and integrate those widths. Pay attention to where the right boundary switches from a parabola to a vertical line, since that means you'll need to split the integral into two parts with different width formulas.

Intuitive explanation

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