To estimate the cross-sectional area of a shallow pond
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
To estimate the cross-sectional area of a shallow pond, environmental scientists measure the water depth along a 20-meter transect perpendicular to the shore. At each position x meters from the shoreline they record the distance (in meters) from the pond’s surface to the muddy bottom. Their depth data are modeled by two functions for 0≤x≤20:
• Surface elevation relative to a fixed datum: y=0 (the x-axis).
• Bottom elevation: b(x)=0.15xsin(10πx)−1.8.
The water occupies the region between the graphs of y=0 and y=b(x), where b(x)<0.
(a) Confirm that b(x)<0 for all 0<x<20, and interpret this inequality in the context of the pond.
(b) Write, but do not yet evaluate, an integral representing the exact water cross-sectional area in square meters.
(c) Evaluate the integral from part (b) analytically to obtain an exact expression for the area.
(d) Compute a numerical approximation of the area to the nearest tenth of a square meter.
(e) Explain how the Fundamental Theorem of Calculus guarantees that the result in part (c) represents the true geometric area of the pond’s cross section.