As viewed from above, a swimming pool has the shape of the ellipse given by
The cross sections perpendicular to the ground and parallel to the -axis are squares. Find the total volume of the pool.
note that the the curve is given by
the plus part gives the curve above the x-axis, the minus part gives the curve below the x-axis.
recall, the volume is given by
where is the cross-sectional area.
note that here, is given by the formula for a square, where the side length is given by the difference between the two functions.
i told you, the height of the square is the difference between the two functions. draw any vertical line in the ellipse that connects the top to the bottom. the length of that line is the side length of the square at that point. it's length is given by the difference of the functions