A rectangular botanical garden contains a decorative reflecting pool

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

A rectangular botanical garden contains a decorative reflecting pool. In a coordinate grid laid over the garden (units in meters), the left edge of the pool follows the curve

x = \left(\frac{y}{2}\right)^2

for

0 \le y \le 6

, while the right edge follows

x = 10 - \left(\frac{y}{3}\right)^2

over the same

y

-interval. The pool is bounded above and below by these two curves at

y = 0

and

y = 6

. (a) Justify which slicing direction—horizontal or vertical—results in a single definite integral for the pool’s surface area. Support your answer by referencing the behavior of the bounding curves. (b) Set up and evaluate the definite integral that yields the exact area of the pool’s surface. (c) Suppose the garden designer wants to widen the pool uniformly by shifting the left edge

0.25\,\text{m}

further left (parallel to the

y

-axis) while keeping the right edge fixed. Without re-deriving the entire integral, determine the new surface area and state the increase in area in exact square meters.

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