A vertical cross-section of a canal is modeled in the coordinate plane as follows
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals
Problem
A vertical cross-section of a canal is modeled in the coordinate plane as follows. The left bank satisfies x=(y+2)2 and the right bank satisfies x=12−0.5(y+2), where x is measured in meters horizontally from an origin on the left bank and y is measured in meters downward from the water surface (y=0 at the surface). The water depth is 4 m (so water occupies 0≤y≤4).
(1) State the interval of y over which water is present in the cross-section.
(2) Decide whether integrating with respect to x or y is more practical for finding the cross-sectional area of water. Justify your choice by referring to the number of definite integrals required and the algebraic form of the integrands.
(3) Using the variable you selected in part (b), set up and evaluate the definite integral that gives the cross-sectional area of water in square meters.
(4) The canal is 200 m long. Using your result from part (c), estimate the volume of water contained in this 200 m stretch, expressing your answer in cubic meters.