Mathematics

Problem

Consider the region in the

xy

-plane enclosed by the curves

x=y^{2}

and

x=8-y

. (a) Determine the

y

-coordinates of the points where the two curves intersect. (b) For each value of

y

on the interval identified in part (a), state which of the two curves forms the right-hand boundary of the enclosed region and justify your conclusion using an appropriate comparison of the two

x

-coordinates. (c) A student proposes to compute the area of the region by evaluating

\displaystyle \int_{y_1}^{y_2}\bigl(y^{2}-(8-y)\bigr)\,dy

, where

y_1

and

y_2

are the intersection values found in part (a). Explain whether the value of this integral will be positive, negative, or zero, and justify your answer in terms of the orientation of left and right boundaries.

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