Two curves in the –plane are given analytically by Curve : Curve : A finite
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Two curves in the xy–plane are given analytically by
Curve P: x=y2
Curve Q: x=6−y
A finite, closed region R is enclosed by these curves.
Answer the following questions about the geometric relationship of the two curves and the region they bound.
(a) Solve analytically for all y–coordinates of the points where curves P and Q intersect.
(b) For each value of y in the open interval determined in part (a), determine which curve constitutes the right‐hand boundary of region R and which constitutes the left‐hand boundary. Fully justify your conclusion without explicitly computing area.
(c) A student proposes to evaluate ∫y1y2(y2−(6−y))dy to obtain the area of R, where y1<y2 are the intersection ordinates found in part (a). Explain in detail why this setup produces a negative result and how the integral should be modified to produce the correct positive area.